diff --git a/-9E0T4oBgHgl3EQfxAH-/content/tmp_files/2301.02642v1.pdf.txt b/-9E0T4oBgHgl3EQfxAH-/content/tmp_files/2301.02642v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2ac2c721f971644edf6486986506be9b1636d1ba --- /dev/null +++ b/-9E0T4oBgHgl3EQfxAH-/content/tmp_files/2301.02642v1.pdf.txt @@ -0,0 +1,931 @@ +Triple-stream Deep Metric Learning of Great Ape Behavioural Actions +Otto Brookes1, Majid Mirmehdi1, Hjalmar K¨uhl2, Tilo Burghardt1 +1Department of Computer Science, University of Bristol, United Kingdom +2Evolutionary and Anthropocene Ecology, iDiv, Leipzig, Germany +otto.brookes@bristol.ac.uk, majid@cs.bris.ac.uk, tilo@cs.bris.ac.uk, hjalmar.kuehl@idiv.de +Keywords: +Animal Biometrics, Multi-stream Deep Metric Learning, Animal Behaviour, Great Apes, PanAf-500 Dataset +Abstract: +We propose the first metric learning system for the recognition of great ape behavioural actions. Our proposed +triple stream embedding architecture works on camera trap videos taken directly in the wild and demonstrates +that the utilisation of an explicit DensePose-C chimpanzee body part segmentation stream effectively com- +plements traditional RGB appearance and optical flow streams. We evaluate system variants with different +feature fusion techniques and long-tail recognition approaches. Results and ablations show performance im- +provements of ∼ 12% in top-1 accuracy over previous results achieved on the PanAf-500 dataset containing +180,000 manually annotated frames across nine behavioural actions. Furthermore, we provide a qualitative +analysis of our findings and augment the metric learning system with long-tail recognition techniques show- +ing that average per class accuracy – critical in the domain – can be improved by ∼ 23% compared to the +literature on that dataset. Finally, since our embedding spaces are constructed as metric, we provide first data- +driven visualisations of the great ape behavioural action spaces revealing emerging geometry and topology. +We hope that the work sparks further interest in this vital application area of computer vision for the benefit of +endangered great apes. We provide all key source code and network weights alongside this publication. +positive +anchor +fusion +negative +ResNet-50 +ResNet-50 +ResNet-50 +x0 + . . . . . . +x128 +x0 + . . . . . . +x128 +x0 + . . . . . . +x128 + Metric +Learning +d +d +d +d +LTriplet +x0 + . . . . . . +x128 +x0 + . . . . . . +x128 +embedding model +DensePose-C +Optical flow +RGB +x0 + . . . . . . +x128 +x0 + . . . . . . +x128 +x0 + . . . . . . +x128 +x0 + . . . . . . +x128 +x0 + . . . . . . +x128 +x0 + . . . . . . +x128 +x0 + . . . . . . +x128 +embedding model +embedding model +shared weights +shared weights +Figure 1: System Overview. Our proposed triple-stream metric learning approach utilises all RGB appearance, optical flow, +and DensePose-C segmentations of chimps in videos. Exploiting hybrid reciprocal triplet and cross entropy losses, the model +is then trained to map embeddings representing great ape behavioural actions onto a metric space, where semantically similar +representations are geometrically close forming natural clusters. This pipeline improves on state-of-the-art classification +performance and allows for visualisations of the underpinning space of behavioural actions. (best viewed zoomed) +1 +INTRODUCTION +As the climate crisis gathers pace, the threat to many +endangered species grows ever more perilous (Al- +mond et al., 2022). All species of great apes are, for +instance, listed as endangered or critically endangered +according to the IUCN Red List (IUCN, 2022) +.. . . . . . . . . . . there is urgent need for methods +Consequently, there is urgent need for methods that +can help to monitor population status and assess the +effectiveness of conservation interventions (K¨uhl and +Burghardt, 2013; Congdon et al., 2022; Tuia et al., +2022). This includes the recognition of behaviors and +variation therein, as an integral part of biological di- +versity (Dominoni et al., 2020; Carvalho et al., 2022). +arXiv:2301.02642v1 [cs.CV] 6 Jan 2023 + +Tripletloss:randomtriplets +camera_interaction +climbing down +climbing_up +hanging +running +sitting +sitting_on_back +standing +walkingTripletloss:randomlyinitialisedweights +camera_interaction +climbing_down +climbing_up +hanging +running +sitting +sitting_on_back +standing +walkingPrevious works have employed deep neural net- +works which leverage multiple modalities, such as +RGB, optical flow, and audio (Sakib and Burghardt, +2020; Bain et al., 2021), for the classification of great +ape behaviours and actions. However, higher level ab- +stractions such as pose or body part information have +remained unexplored for addressing this task. In re- +sponse, we propose utilising the latter together with +RGB and optical flow in a triple-stream metric learn- +ing system (see Fig. 1) for improved classification re- +sults and domain visualisations relevant to biologists. +104 +105 +# Samples (log) +Behavioural action classes +hanging +walking +sitting on back +standing +sitting +climbing up +camera interaction +running +climbing down +Figure 2: Behavioural Actions in the PanAf-500 Data. +Examples of each one of the nine behavioural action classes +(top) and their distribution across the approx. 180k frames +in the dataset (bottom). Note the imbalance of two orders of +magnitude in the distribution. (best viewed zoomed) +Great Ape Activities - This paper will focus on +great ape activity recognition, where the coarse ac- +tivity classes used are illustrated in Fig. 2 for the +utilised PanAf-500 dataset (see Sec. 3). +Note that +computer vision would traditionally categorise these +classes as actions whilst in the biological realm they +represent behaviour (or aspects thereof) often cap- +tured in ethograms (Nishida et al., 1999; Zamma and +Matsusaka, 2015). For clarity, in this paper we will +refer to these classes as behavioural actions recognis- +ing historical traditions in both disciplines. +We will approach the classification task via a deep +metric learning system (Karaderi et al., 2022) that +embeds inputs into a latent space and uses geomet- +ric distances to form distributions that align with the +semantic similarity captured by the classes (Hermans +et al., 2017; Musgrave et al., 2020). A major advan- +tage over standard supervised systems is that sample +distances in visualisations of the latent space always +relate to learned similarity and, thus, are more natu- +rally interpretable by experts. We will also analyse +the role that additional DensePose-Chimp informa- +tion (Sanakoyeu et al., 2020) can play in improving +recognition performance compared to systems that +utilise RGB and optical flow only. Lastly, as shown +by Sakib and Burghardt (Sakib and Burghardt, 2020), +there are significant challenges in correctly classify- +ing behavioural actions which occur infrequently and +form the distribution tail (see Fig. 2). To address this, +we will employ three long-tailed recognition (LTR) +techniques to improve performance on tail classes; (i) +logit adjustment (Menon et al., 2020); (ii) class bal- +anced focal loss (Cui et al., 2019); and (iii) weight +balancing (Alshammari et al., 2022). +In summary, our contributions are as follows: +(i) we implement the first deep metric learning system +for recognising great ape behavioural actions; (ii) we +show that utilising explicit pose information has a sig- +nificant positive effect on recognition performance in +this domain; and (iii) we establish that existing LTR +techniques can be applied in a metric learning setting +to improve performance on tail classes for the prob- +lem. The proposed approaches improve the state-of- +the-art performance benchmarks with respect to top-1 +(∼ 85%) and average per class (∼ 65%) accuracy on +the PanAf-500 dataset. +2 +RELATED WORK +Action recognition aims to classify actions observed +in video (Kalfaoglu et al., 2020; Shaikh and Chai, +2021). Learning spatio-temporal features character- +istic for actions (Simonyan and Zisserman, 2014) via +various deep learning paradigms forms the approach +of choice in the domain of human action recogni- +tion (HAR). We will briefly review concepts from this +field, before discussing specifc relevant great ape be- +havioural action recognition and LTR methods. +Human Action Recognition - Although there are +numerous deep learning approaches to action recog- +nition (Zhou et al., 2018; Lin et al., 2019; Tran et al., +2019; Kalfaoglu et al., 2020; Pan et al., 2019; Majd +and Safabakhsh, 2020; Sharir et al., 2021; Zhang +et al., 2021a) this work focuses on multi-stream ar- +chitectures, which address key aspects of the action +recognition problem (e.g., spatial and temporal) in- + +dependently and explicitly. Feichtenhofer et al. (Fe- +ichtenhofer et al., 2019) introduced the SlowFast ar- +chitecture which employs two streams, each operat- +ing at different frame rates; a slow, low frame-rate +pathway captures spatial information while the fast, +high frame-rate pathway captures fine temporal detail. +Other types of multi-stream networks process differ- +ent visual modalities. Simonyan (Simonyan and Zis- +serman, 2014) introduced a two-stream network that +processes RGB and optical flow to exploit spatial and +temporal semantics, respectively. Since then, several +networks that utilise additional modalities, such as +motion saliency (Zong et al., 2021) and audio (Wang +et al., 2021), have been introduced. Recently, the in- +troduction of pose, which is critical for the perception +of actions (Le et al., 2022), has shown promising re- +sults in multi-stream architectures (Hong et al., 2019; +Hayakawa and Dariush, 2020; Duan et al., 2021; Li +et al., 2022). +In particular, the DensePose format +provides an opportunity to exploit fine-grained, seg- +mentation map-based pose representations for action +recognition. Hayakawa et al. (Hayakawa and Dar- +iush, 2020) combine RGB and DensePose estimations +in a two-stream network and demonstrate strong per- +formance on egocentric footage of humans. Whilst +such significant progress has been made in the domain +of HAR, research into great ape behavioural action +recognition is still in its infancy and few systems have +been tested on natural datasets. +Great Ape Domain - +To date, two systems +have attempted automated great ape behavioural ac- +tion recognition, both are multi-stream architectures. +The first (Sakib and Burghardt, 2020) is based on the +two-stream convolutional architecture by Simonyan +et al. (Simonyan and Zisserman, 2014) and used 3D +ResNet-18s for feature extraction and LSTM-based +fusion of RGB and optical flow features. They report +top-1 accuracy of 73.52% across the nine behavioural +actions in the PanAf-500 dataset (see Sec. 3) and a +relatively low average per class accuracy (42.33%), +highlighting the issue of tail class performance. The +second, proposed by Bain et al. (Bain et al., 2021), +is a deep learning system that requires both audio +and video inputs and detects two specific behaviours; +buttress drumming and nut cracking. Their system +utilised a 3D ResNet-18 and a 2D ResNet-18 for ex- +traction of visual and assisting audio features, respec- +tively, in different streams. They achieved an aver- +age precision of 87% for buttress drumming and 85% +for nut cracking on their unpublished dataset. How- +ever, the multi-modal method is not applicable to all +camera trap settings since many older models do not +provide audio. It cannot be utilised on the PanAf-500 +dataset since many clips there do not contain audio. +Long-tailed Recognition - Most natural recorded +data exhibits long-tailed class distributions (Liu et al., +2019). This is true of great ape camera-trap footage +which is dominated by commonly occurring be- +haviours - even with only the nine classes of the +PanAf-500 data the distribution shows a clear tail (see +Fig. 2). Without addressing this issue, models trained +on such data often exhibit poor performance on rare +classes. +Various counter-measures have been pro- +posed (Verma et al., 2018; Kang et al., 2019; Zhang +et al., 2021b). +Class balanced losses assign addi- +tional weights, typically determined by inverse class +frequencies, to samples from rare classes and have +yielded strong results when coupled with techniques +to reduce per-class redundancy (Cui et al., 2019). +Similarly, logit adjustment uses class frequencies to +directly offset output logits in favour of minority +classes during training (Menon et al., 2020). +An +orthogonal approach, based on the observation that +weight norms for rare classes are smaller in naively +trained classifiers, is to perform weight balancing (Al- +shammari et al., 2022). +These techniques have +achieved strong results on several LTR benchmarks. +Before detailing how we use triple-stream metric +learning with explicit DensePose-Chimp processing +and LTR extensions for behavioural action recogni- +tion, we will briefly outline the utilised dataset. +3 +DATASET +The Pan-African dataset, gathered by the Pan African +Programme: ‘The Cultured Chimpanzee’, comprises +∼ 20,000 videos from footage gathered at 39 study +sites spanning 15 African countries. Here we utilise +a 500 video subset, PanAf-500, specifically ground- +truth labelled for use in computer vision under re- +producible and comparable benchmarks. It includes +frame-by-frame annotations for full-body locations of +great apes and nine behavioural actions (Sakib and +Burghardt, 2020) across approximately 180k frames +(see. Fig. 3). Fig. 2 displays the behavioural actions +classes in focus together with their distribution. We +utilised the PanAf-500 dataset for all experiments and +employ the same training and test partitions described +in (Sakib and Burghardt, 2020). +4 +METHOD +The proposed system utilises three visual modali- +ties as input; RGB, optical flow, and DensePose-C +estimations (Sanakoyeu et al., 2020), as illustrated +in Fig. 1). All optical flow images are pre-computed +using OpenCV’s implementation of the Dual TV L1 +algorithm (Zach et al., 2007). We employ the model +developed by Sanakoyeu et al. (Sanakoyeu et al., + +Figure 3: Frame-by-frame Ground Truth Annotations. +Four still frames from PanAf-500 videos with annotations +of location (green boxes) and behavioural actions (visu- +alised as text) of the apes in-frame. (best viewed zoomed) +x0 .... x128 +Concatenation +Conv3D +ResNet50 +ResNet50 +ResNet50 +b x 6144 x 5 x 8 x 8 +MaxPool3D +Feature maps +2048 x 5 x 8 x 8 +feature extractors +Triple stream +AdaptiveAvgPool3D +Fc1 +b x 2048 x 3 x 5 x 5 +Fc2 +b x 1024 +x0 .... x128 +x0 .... x128 +x0 .... x128 +Multiplication +L2 norm +x0 .... x128 +RGB +Optical flow +Output +DensePose-C +Figure 4: Fusion Head Schematics. A component break- +down of fusion by element-wise multiplication (left) and +convolutional fusion (right) as applied for our work to ex- +plore their impact on performance. +2020) to generate DensePose-C segmentations de- +scribing chimpanzee pose. The model predicts dense +correspondences between image pixels and a 3-D ob- +ject mesh where each mesh represents a chimpanzee +body part specified by a selector I and local surface +coordinates within each mesh indexed by U and V. +Frame-by-frame application to each of the PanAf- +500 videos yields DensePose-C estimates expressed +in IUV coordinates. +Each of the three input modalities is fed into a 3D +ResNet-50 (Du Tran et al., 2017) backbone, which +together act as a feature extractor (see Fig. 1). The +input tensors into the backbones are 3D since inputs +are processed in snippets, that is each stream accepts a +sequence of n consecutive RGB frames, optical flow +images, or IUV coordinates, respectively. The final +fully-connected layer outputs an n-dimensional en- +coding for each stream. These are fused into a single +embedding using three popular approaches; (i) sim- +ple averaging across streams; (ii) convolutional fusion +whereby stream features are concatenated and passed +to a 3D convolutional layer as a volume; and (iii) +element-wise multiplication of all three embedding +vectors followed by L2 normalisation. The latter two +approaches are illustrated in detail in Fig. 4. A lin- +ear layer at the end of the fusion head finally outputs +the unified embedding as logits. Whilst this system +was trained via metric learning - visually sketched in +Fig. 1 (right) - a k-NN classifier is used to perform +inference in the embedding space during evaluation. +Let the parameters of this network fθ(·) be de- +noted by θ. Furthermore, let fθ(x) = x be the short- +hand for referring to embeddings. Our metric learn- +ing objective is, thus, to minimise the distance be- +tween anchor-positive embedding pairs d(xa,xp) and +maximise distance between anchor-negative embed- +ding pairs d(xa,xn), where d represents a Euclidean. +Instead of using standard triplet loss (Hermans et al., +2017) LTL, we use an improved version (Andrew +et al., 2021), where the model is optimised via a hy- +brid reciprocal triplet and softmax cross-entropy loss: +LRC = LCE +λ LRT. +(1) +It is assembled from two components balanced by +λ = 0.1 as given in (Andrew et al., 2021). The two +components themselves are evaluated as: +LRT = d(xa,xp)+ +1 +d(xa,xn) +(2) +LCE = −log +� +exy +∑C +i=1 exi +� +, +(3) +where C denotes the total number of classes and y are +the class labels. +In order to extend this system into the LTR do- +main we substitute the softmax cross-entropy term +for losses calculated using; (i) cross-entropy soft- +max with logit adjustment (Menon et al., 2020) LLA; +(ii) class-balanced focal loss (Cui et al., 2019) LCB; +and (iii) class-balanced focal loss with weight balanc- +ing (Alshammari et al., 2022). The first two losses are +evaluated as follows: +LLA = −log +� exy +τ · log πy +∑C +i=1 exi+τ · log πi +� +, +(4) +LCB = − 1−β +1−βny +C +∑ +i=1 +(1− pi)γ log(pi), +(5) + +Bushnestandingcomera_interaction +camara +interactionwhere π represents the class priors (i.e., class frequen- +cies in the training set) and temperature factor τ = 1, +β = 0.99 is the re-weighting hyper-parameter, n is the +total number of samples, y are the classes, γ = 1 is the +focal loss hyper-parameter and pi = σ(xi). Balancing +the network weights θ is performed via a MaxNorm +constraint ∥θl,i∥2 +2 ≤ δ2,∀i given in (Alshammari et al., +2022) imposed on each class filter i in the last layer l +of the network where δ is the L2-norm ball radius. We +will reference a LCB-based optimisation where weight +balancing is performed via LWB. +Methodologically, this described architecture ap- +proaches the learning of behavioural great ape actions +via five key capabilities: 1) utilisation of multiple rel- +evant input modalities across an entire video snippet; +2) effective streamed content encoding; 3) fusion into +a single embedding space; 4) metric space optimisa- +tion so that distances naturally reflect semantic sim- +ilarity; and 5) taking into account class imbalances +common to the domain content. +5 +EXPERIMENTS +5.1 +General Training Setup +We train our architecture via SGD optimisation using +batch size 32 and learning rate 10−4. Feature extrac- +tor backbones are initialised with Kinetics-400 (Kay +et al., 2017) pre-trained weights and training runs are +distributed over 8 Tesla V100 GPUs for 100 epochs. +5.2 +Baselines and Stream Ablations +As shown in Tab. 1, we first establish performance +benchmarks for one and two stream baseline archi- +tectures of our system (rows 2–5) against the cur- +rent state-of-the-art (row 1), which uses a ResNet-18 +backbone with focal loss LFL, SGD, and LSTM-based +frame fusion (Sakib and Burghardt, 2020). As ex- +pected, we confirmed that - using identical setups and +losses - adding an optical flow stream is beneficial +in the great ape domain mirroring HAR results (see +rows 2 vs 4, and 3 vs 5). Additionally, models trained +using LRC consistently outperformed standard triplet +loss LRC scenarios (see rows 2 vs 3, and 4 vs 5). Fi- +nally, a dual-stream version of our proposed architec- +ture trained with LRC outperforms the state-of-the-art +by a small margin (see rows 1 vs 5). +5.3 +Triple-Stream Recognition +As given in Tab. 1 rows 6–8, our proposed triple- +stream architecture significantly outperforms all base- +lines with regards to top-1 accuracy, achieving up to +85.86%. Thus, explicit DensePose-C information ap- +pears a useful information source for boosting be- +havioural action recognition in great apes. However, +Table 1: Behavioural Action Recognition Benchmarks. +Top-1 and average per-class (C-Avg) accuracy performance +on the PanAf-500 dataset for the current state-of-the- +art (row 1), single and dual-stream baselines (rows 2–5), +and our triple-stream networks (rows 6–8) for different fu- +sion methodologies and losses tested. +Models/Streams +Fusion +Loss +Top-1 +C-Avg +Sakib et al. 2020 +1 +RGB+OF +LSTM +LFL +73.52% +42.33% +Up to Dual-Stream +2 +RGB only +None +LTL +55.50% +32.67% +3 +RGB only +None +LRC +74.24% +55.76% +4 +RGB+OF +Avg +LTL +62.90% +39.10% +5 +RGB+OF +Avg +LRC +75.02% +61.97% +Triple-Stream (Ours) +6 +RGB+OF+DP +Avg +LRC +81.71% +46.61% +7 +RGB+OF+DP +Conv +LRC +82.04% +56.31% +8 +RGB+OF+DP +Elem +LRC +85.86% +50.50% +without LTR techniques all our triple-stream models +are significantly outperformed by a dual-stream set- +ting (row 5) with regards to average per-class accu- +racy. This reduction is caused by significantly poorer +performance on minority classes (see Sec. 5.4). +Since the learned behavioural action embeddings +are constructed as metric from the outset, they can +be visualised meaningfully – we note that such data- +driven visualisations are novel in the primatology do- +main. Fig. 5 depicts such learned spaces for our data +and architecture where, independent of stream cardi- +nality, embeddings cluster the training data cleanly. +This is of course expected given above 99% top-1 +training accuracy in all settings. Yet, behavioural ac- +tions of great apes are highly intricate as well as vari- +able and, even with approx. 144,000 training frames +used, the model clearly shows signs of overfitting. As +a result, test set embeddings exhibit significant cluster +overlap. Sample groups representing sitting, standing, +and walking, for instance, blend into one another. In +addition to overfitting, this also highlights the transi- +tional nature of these often temporarily adjacent and +smoothly changing actions. Thus, future temporally +transitional ground truth labelling may be needed to +represent behavioural great ape action in the PanAf- +500 dataset more authentically. +5.4 +Fusing Streams +When looking at the impact of information fusion +methods on performance in more detail, we find that +benchmarks vary significantly (see Tab. 1 rows 6–8) +when we test averaging, element-wise multiplication, +and convolutional fusion, as described in Sec. 4. Re- +sults show that convolution and element-wise mul- +tiplication improve performance slightly across both +metrics when compared with averaging: top-1 accu- + +camera interaction +climbing up +climbing down +hanging +running +sitting +sitting on back +standing +walking +Behavioural actions +Single Stream (RGB) +Kinetics pretrained (no training) +Training +Training +Test +Dual Stream (RGB+OF) +Triple Stream (AllThree) +Figure 5: Visualisations of Great Ape Behavioural Action Spaces. A 2D t-SNE (Wattenberg et al., 2016) visualisation of +the 128-dimensional training (top-right) and test (bottom-right) embeddings produced by the single, dual and three-stream +network with convolutional fusion. We can see that training set embeddings from all classes are clustered cleanly. In contrast, +test set embeddings show significant overlap and only embeddings from majority classes form distinct clusters. This is +consistent with the high top-1 accuracy and relatively low average per-class accuracy reported in Tab. 1 +racy improves by 0.33% and 4.1%, respectively (see +rows 6–8). However, the most significant gains are +observed with respect to average per class accuracy +which increases by 3.44% for element-wise multipli- +cation and 9.7% for convolutional fusion. Learnable +parameters in the convolution method clearly help +blending information even when only fewer samples +are available for training. Building on this improve- +ment, we will next investigate the impact of LTR +methods in order to benefit tail class performance. +5.5 +Long-tail Recognition +When grouping behavioural actions into head (cov- +ering sitting, standing, and walking) and remain- +ing tail classes based on frequency in the data (see +Fig. 2), a significant performance gap becomes appar- +ent even when using the so far best C-Avg performing +model (see Tab. 2 row 1). Employing LTR techniques +can, however, reduce this gap and improve average +per-class accuracy further as quantified across rows +2–4 in Tab. 2). Fig. 6 shows t-SNE visualisations of +the three LTR triple-stream approaches when trained +with convolutional feature fusion. Particularly for the +class-balanced approaches and weight-balancing se- +tups (two rightmost), tail class clusters appear more +clearly separated and class overlap is generally re- +duced. Thus, for the great ape domain underrepre- +sented classes are indeed an effective source of infor- +mation for improving action separability in general. +6 +CONCLUSION +In this work we introduced the first deep metric learn- +ing system for great ape behavioural action recogni- +tion. We demonstrated that the proposed triple-stream +architecture can provide leading state-of-the-art per- +formance when tested on the PanAf-500 camera trap +dataset covering 180,000 annotated frames across 500 +videos taken in the wild. We demonstrated that the ad- +dition of a DensePose-C chimpanzee pose estimation +stream into the embedding architecture is highly ef- +fective and leads to system performance of 85.86% +top-1 accuracy on the data. +We also showed that +adding LTR techniques that address poor tail class +performance to the system can improve the average +per-class accuracy to 65.66% on the dataset. Despite +these improvements we note that both larger anno- +tated datasets to counteract overfitting as well as more +temporally blended forms of annotation (e.g. action +transition annotations) would benefit the authenticity +of data-driven great ape behavioural representations. +We hope that the research presented here sparks fur- +ther interest in this vital application area for the bene- +fit of endangered species such as great apes. +ACKNOWLEDGEMENTS +We thank the Pan African Programme: +‘The Cultured +Chimpanzee’ team and its collaborators for allowing the use +of their data for this paper. We thank Amelie Pettrich, An- +tonio Buzharevski, Eva Martinez Garcia, Ivana Kirchmair, + +climbing up +hanging +running +sitting +sitting on back +standing +walking +Logit adjustment +No LTR augmentation +Weight balanced +CB (+focal loss) +climbing down +camera interaction +Test +Figure 6: Long-tail Test Embeddings. A 2D t-SNE visualisation of the 128-dimensional test embeddings produced by the +three-stream network with convolutional fusion alone (leftmost) and augmented with each LTR technique; (i) logit adjustment +(ii) CB (+focal loss) and (iii) weight balancing. All LTR-augmented methods improve clustering of embeddings belonging to +tail classes. They appear more clearly separated and exhibit less overlap when compared with the non-LTR method. +Sebastian Sch¨utte, Linda Gerlach and Fabina Haas. We also +thank management and support staff across all sites; specif- +ically Yasmin Moebius, Geoffrey Muhanguzi, Martha Rob- +bins, Henk Eshuis, Sergio Marrocoli and John Hart. Thanks +to the team at https://www.chimpandsee.org particularly +Briana Harder, Anja Landsmann, Laura K. Lynn, Zuzana +Mach´aˇckov´a, Heidi Pfund, Kristeena Sigler and Jane Wid- +ness. The work that allowed for the collection of the dataset +was funded by the Max Planck Society, Max Planck Society +Innovation Fund, and Heinz L. Krekeler. In this respect we +would like to thank: Ministre des Eaux et Forˆets, Minist`ere +de l’Enseignement sup´erieur et de la Recherche scientifique +in Cˆote d’Ivoire; Institut Congolais pour la Conservation de +la Nature, Minist`ere de la Recherche Scientifique in Demo- +cratic Republic of Congo; Forestry Development Authority +in Liberia; Direction Des Eaux Et Forˆets, Chasses Et Con- +servation Des Sols in Senegal; Makerere University Biolog- +ical Field Station, Uganda National Council for Science and +Technology, Uganda Wildlife Authority, National Forestry +Authority in Uganda; National Institute for Forestry De- +velopment and Protected Area Management, Ministry of +Agriculture and Forests, Ministry of Fisheries and Environ- +ment in Equatorial Guinea. This work was supported by the +UKRI CDT in Interactive AI under grant EP/S022937/1. +Table 2: +LTR-enabled Behavioural Action Recogni- +tion Benchmarks. +Average per-class accuracy for our +triple-stream network with convolutional fusion for best +performing non-LTR method (row1), and three LTR ap- +proaches (rows 2–4) targetting poor tail class performance. +Method/Loss +C-Avg +Head +Tail +Non-LTR Triple-Stream +1 +LRC +56.31 +80.57 +44.78 +LTR Triple-Stream +2 +LLA +61.76 +83.22 +50.7 +3 +LCB +63.56 +77.60 +55.95 +4 +LWB +65.66 +82.55 +56.26 +REFERENCES +Almond, R., Grooten, M., Juffe Bignoli, D., and Petersen, +T. (2022). +Wwf (2022) living planet report 2022 - +building a nature-positive society. 1 +Alshammari, S., Wang, Y.-X., Ramanan, D., and Kong, S. +(2022). Long-tailed recognition via weight balancing. +In CVPR, pages 6897–6907. 2, 3, 4, 5 +Andrew, W., Gao, J., Mullan, S., Campbell, N., Dowsey, +A. W., and Burghardt, T. (2021). 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Image and Vision Com- +puting, 107:104108. 3 + diff --git a/-9E0T4oBgHgl3EQfxAH-/content/tmp_files/load_file.txt b/-9E0T4oBgHgl3EQfxAH-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f454bb3c17c2eb7d946730e0281b4d617c513e98 --- /dev/null +++ b/-9E0T4oBgHgl3EQfxAH-/content/tmp_files/load_file.txt @@ -0,0 +1,829 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf,len=828 +page_content='Triple-stream Deep Metric Learning of Great Ape Behavioural Actions Otto Brookes1, Majid Mirmehdi1, Hjalmar K¨uhl2, Tilo Burghardt1 1Department of Computer Science, University of Bristol, United Kingdom 2Evolutionary and Anthropocene Ecology, iDiv, Leipzig, Germany otto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='brookes@bristol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='uk, majid@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='bris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='uk, tilo@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='bris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='uk, hjalmar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='kuehl@idiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='de Keywords: Animal Biometrics, Multi-stream Deep Metric Learning, Animal Behaviour, Great Apes, PanAf-500 Dataset Abstract: We propose the first metric learning system for the recognition of great ape behavioural actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Our proposed triple stream embedding architecture works on camera trap videos taken directly in the wild and demonstrates that the utilisation of an explicit DensePose-C chimpanzee body part segmentation stream effectively com- plements traditional RGB appearance and optical flow streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' We evaluate system variants with different feature fusion techniques and long-tail recognition approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Results and ablations show performance im- provements of ∼ 12% in top-1 accuracy over previous results achieved on the PanAf-500 dataset containing 180,000 manually annotated frames across nine behavioural actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Furthermore, we provide a qualitative analysis of our findings and augment the metric learning system with long-tail recognition techniques show- ing that average per class accuracy – critical in the domain – can be improved by ∼ 23% compared to the literature on that dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Finally, since our embedding spaces are constructed as metric, we provide first data- driven visualisations of the great ape behavioural action spaces revealing emerging geometry and topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' We hope that the work sparks further interest in this vital application area of computer vision for the benefit of endangered great apes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' We provide all key source code and network weights alongside this publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' positive anchor fusion negative ResNet-50 ResNet-50 ResNet-50 x0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' x128 x0 .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' x128 embedding model embedding model shared weights shared weights Figure 1: System Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Our proposed triple-stream metric learning approach utilises all RGB appearance, optical flow, and DensePose-C segmentations of chimps in videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Exploiting hybrid reciprocal triplet and cross entropy losses, the model is then trained to map embeddings representing great ape behavioural actions onto a metric space, where semantically similar representations are geometrically close forming natural clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' This pipeline improves on state-of-the-art classification performance and allows for visualisations of the underpinning space of behavioural actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' (best viewed zoomed) 1 INTRODUCTION As the climate crisis gathers pace, the threat to many endangered species grows ever more perilous (Al- mond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' All species of great apes are, for instance, listed as endangered or critically endangered according to the IUCN Red List (IUCN, 2022) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' there is urgent need for methods Consequently, there is urgent need for methods that can help to monitor population status and assess the effectiveness of conservation interventions (K¨uhl and Burghardt, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Congdon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Tuia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' This includes the recognition of behaviors and variation therein, as an integral part of biological di- versity (Dominoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Carvalho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='02642v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='CV] 6 Jan 2023 Tripletloss:randomtriplets camera_interaction climbing down climbing_up hanging running sitting sitting_on_back standing walkingTripletloss:randomlyinitialisedweights camera_interaction climbing_down climbing_up hanging running sitting sitting_on_back standing walkingPrevious works have employed deep neural net- works which leverage multiple modalities, such as RGB, optical flow, and audio (Sakib and Burghardt, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Bain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2021), for the classification of great ape behaviours and actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' However, higher level ab- stractions such as pose or body part information have remained unexplored for addressing this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' In re- sponse, we propose utilising the latter together with RGB and optical flow in a triple-stream metric learn- ing system (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 1) for improved classification re- sults and domain visualisations relevant to biologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 104 105 # Samples (log) Behavioural action classes hanging walking sitting on back standing sitting climbing up camera interaction running climbing down Figure 2: Behavioural Actions in the PanAf-500 Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Examples of each one of the nine behavioural action classes (top) and their distribution across the approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 180k frames in the dataset (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Note the imbalance of two orders of magnitude in the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' (best viewed zoomed) Great Ape Activities - This paper will focus on great ape activity recognition, where the coarse ac- tivity classes used are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 2 for the utilised PanAf-500 dataset (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Note that computer vision would traditionally categorise these classes as actions whilst in the biological realm they represent behaviour (or aspects thereof) often cap- tured in ethograms (Nishida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Zamma and Matsusaka, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' For clarity, in this paper we will refer to these classes as behavioural actions recognis- ing historical traditions in both disciplines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' We will approach the classification task via a deep metric learning system (Karaderi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2022) that embeds inputs into a latent space and uses geomet- ric distances to form distributions that align with the semantic similarity captured by the classes (Hermans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Musgrave et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' A major advan- tage over standard supervised systems is that sample distances in visualisations of the latent space always relate to learned similarity and, thus, are more natu- rally interpretable by experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' We will also analyse the role that additional DensePose-Chimp informa- tion (Sanakoyeu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2020) can play in improving recognition performance compared to systems that utilise RGB and optical flow only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Lastly, as shown by Sakib and Burghardt (Sakib and Burghardt, 2020), there are significant challenges in correctly classify- ing behavioural actions which occur infrequently and form the distribution tail (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' To address this, we will employ three long-tailed recognition (LTR) techniques to improve performance on tail classes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' (i) logit adjustment (Menon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' (ii) class bal- anced focal loss (Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' and (iii) weight balancing (Alshammari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' In summary, our contributions are as follows: (i) we implement the first deep metric learning system for recognising great ape behavioural actions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' (ii) we show that utilising explicit pose information has a sig- nificant positive effect on recognition performance in this domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' and (iii) we establish that existing LTR techniques can be applied in a metric learning setting to improve performance on tail classes for the prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' The proposed approaches improve the state-of- the-art performance benchmarks with respect to top-1 (∼ 85%) and average per class (∼ 65%) accuracy on the PanAf-500 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 2 RELATED WORK Action recognition aims to classify actions observed in video (Kalfaoglu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Shaikh and Chai, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Learning spatio-temporal features character- istic for actions (Simonyan and Zisserman, 2014) via various deep learning paradigms forms the approach of choice in the domain of human action recogni- tion (HAR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' We will briefly review concepts from this field, before discussing specifc relevant great ape be- havioural action recognition and LTR methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Human Action Recognition - Although there are numerous deep learning approaches to action recog- nition (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Tran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Kalfaoglu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Majd and Safabakhsh, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Sharir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2021a) this work focuses on multi-stream ar- chitectures, which address key aspects of the action recognition problem (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', spatial and temporal) in- dependently and explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Feichtenhofer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' (Fe- ichtenhofer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2019) introduced the SlowFast ar- chitecture which employs two streams, each operat- ing at different frame rates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' a slow, low frame-rate pathway captures spatial information while the fast, high frame-rate pathway captures fine temporal detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Other types of multi-stream networks process differ- ent visual modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Simonyan (Simonyan and Zis- serman, 2014) introduced a two-stream network that processes RGB and optical flow to exploit spatial and temporal semantics, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Since then, several networks that utilise additional modalities, such as motion saliency (Zong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2021) and audio (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2021), have been introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Recently, the in- troduction of pose, which is critical for the perception of actions (Le et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2022), has shown promising re- sults in multi-stream architectures (Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Hayakawa and Dariush, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' In particular, the DensePose format provides an opportunity to exploit fine-grained, seg- mentation map-based pose representations for action recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Hayakawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' (Hayakawa and Dar- iush, 2020) combine RGB and DensePose estimations in a two-stream network and demonstrate strong per- formance on egocentric footage of humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Whilst such significant progress has been made in the domain of HAR, research into great ape behavioural action recognition is still in its infancy and few systems have been tested on natural datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Great Ape Domain - To date, two systems have attempted automated great ape behavioural ac- tion recognition, both are multi-stream architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' The first (Sakib and Burghardt, 2020) is based on the two-stream convolutional architecture by Simonyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' (Simonyan and Zisserman, 2014) and used 3D ResNet-18s for feature extraction and LSTM-based fusion of RGB and optical flow features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' They report top-1 accuracy of 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='52% across the nine behavioural actions in the PanAf-500 dataset (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 3) and a relatively low average per class accuracy (42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='33%), highlighting the issue of tail class performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' The second, proposed by Bain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' (Bain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2021), is a deep learning system that requires both audio and video inputs and detects two specific behaviours;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' buttress drumming and nut cracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Their system utilised a 3D ResNet-18 and a 2D ResNet-18 for ex- traction of visual and assisting audio features, respec- tively, in different streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' They achieved an aver- age precision of 87% for buttress drumming and 85% for nut cracking on their unpublished dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' How- ever, the multi-modal method is not applicable to all camera trap settings since many older models do not provide audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' It cannot be utilised on the PanAf-500 dataset since many clips there do not contain audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Long-tailed Recognition - Most natural recorded data exhibits long-tailed class distributions (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' This is true of great ape camera-trap footage which is dominated by commonly occurring be- haviours - even with only the nine classes of the PanAf-500 data the distribution shows a clear tail (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Without addressing this issue, models trained on such data often exhibit poor performance on rare classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Various counter-measures have been pro- posed (Verma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Class balanced losses assign addi- tional weights, typically determined by inverse class frequencies, to samples from rare classes and have yielded strong results when coupled with techniques to reduce per-class redundancy (Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Similarly, logit adjustment uses class frequencies to directly offset output logits in favour of minority classes during training (Menon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' An orthogonal approach, based on the observation that weight norms for rare classes are smaller in naively trained classifiers, is to perform weight balancing (Al- shammari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' These techniques have achieved strong results on several LTR benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Before detailing how we use triple-stream metric learning with explicit DensePose-Chimp processing and LTR extensions for behavioural action recogni- tion, we will briefly outline the utilised dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 3 DATASET The Pan-African dataset, gathered by the Pan African Programme: ‘The Cultured Chimpanzee’, comprises ∼ 20,000 videos from footage gathered at 39 study sites spanning 15 African countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Here we utilise a 500 video subset, PanAf-500, specifically ground- truth labelled for use in computer vision under re- producible and comparable benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' It includes frame-by-frame annotations for full-body locations of great apes and nine behavioural actions (Sakib and Burghardt, 2020) across approximately 180k frames (see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 2 displays the behavioural actions classes in focus together with their distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' We utilised the PanAf-500 dataset for all experiments and employ the same training and test partitions described in (Sakib and Burghardt, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 4 METHOD The proposed system utilises three visual modali- ties as input;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' RGB, optical flow, and DensePose-C estimations (Sanakoyeu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2020), as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' All optical flow images are pre-computed using OpenCV’s implementation of the Dual TV L1 algorithm (Zach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' We employ the model developed by Sanakoyeu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' (Sanakoyeu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', Figure 3: Frame-by-frame Ground Truth Annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Four still frames from PanAf-500 videos with annotations of location (green boxes) and behavioural actions (visu- alised as text) of the apes in-frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' (best viewed zoomed) x0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='. x128 Concatenation Conv3D ResNet50 ResNet50 ResNet50 b x 6144 x 5 x 8 x 8 MaxPool3D Feature maps 2048 x 5 x 8 x 8 feature extractors Triple stream AdaptiveAvgPool3D Fc1 b x 2048 x 3 x 5 x 5 Fc2 b x 1024 x0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='. x128 x0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='. x128 x0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='. x128 Multiplication L2 norm x0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='. x128 RGB Optical flow Output DensePose-C Figure 4: Fusion Head Schematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' A component break- down of fusion by element-wise multiplication (left) and convolutional fusion (right) as applied for our work to ex- plore their impact on performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 2020) to generate DensePose-C segmentations de- scribing chimpanzee pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' The model predicts dense correspondences between image pixels and a 3-D ob- ject mesh where each mesh represents a chimpanzee body part specified by a selector I and local surface coordinates within each mesh indexed by U and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Frame-by-frame application to each of the PanAf- 500 videos yields DensePose-C estimates expressed in IUV coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Each of the three input modalities is fed into a 3D ResNet-50 (Du Tran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2017) backbone, which together act as a feature extractor (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' The input tensors into the backbones are 3D since inputs are processed in snippets, that is each stream accepts a sequence of n consecutive RGB frames, optical flow images, or IUV coordinates, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' The final fully-connected layer outputs an n-dimensional en- coding for each stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' These are fused into a single embedding using three popular approaches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' (i) sim- ple averaging across streams;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' (ii) convolutional fusion whereby stream features are concatenated and passed to a 3D convolutional layer as a volume;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' and (iii) element-wise multiplication of all three embedding vectors followed by L2 normalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' The latter two approaches are illustrated in detail in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' A lin- ear layer at the end of the fusion head finally outputs the unified embedding as logits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Whilst this system was trained via metric learning - visually sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 1 (right) - a k-NN classifier is used to perform inference in the embedding space during evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Let the parameters of this network fθ(·) be de- noted by θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Furthermore, let fθ(x) = x be the short- hand for referring to embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Our metric learn- ing objective is, thus, to minimise the distance be- tween anchor-positive embedding pairs d(xa,xp) and maximise distance between anchor-negative embed- ding pairs d(xa,xn), where d represents a Euclidean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Instead of using standard triplet loss (Hermans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2017) LTL, we use an improved version (Andrew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2021), where the model is optimised via a hy- brid reciprocal triplet and softmax cross-entropy loss: LRC = LCE +λ LRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' (1) It is assembled from two components balanced by λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='1 as given in (Andrew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' The two components themselves are evaluated as: LRT = d(xa,xp)+ 1 d(xa,xn) (2) LCE = −log � exy ∑C i=1 exi � , (3) where C denotes the total number of classes and y are the class labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' In order to extend this system into the LTR do- main we substitute the softmax cross-entropy term for losses calculated using;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' (i) cross-entropy soft- max with logit adjustment (Menon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2020) LLA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' (ii) class-balanced focal loss (Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2019) LCB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' and (iii) class-balanced focal loss with weight balanc- ing (Alshammari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' The first two losses are evaluated as follows: LLA = −log � exy +τ · log πy ∑C i=1 exi+τ · log πi � , (4) LCB = − 1−β 1−βny C ∑ i=1 (1− pi)γ log(pi), (5) Bushnestandingcomera_interaction camara interactionwhere π represents the class priors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', class frequen- cies in the training set) and temperature factor τ = 1, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='99 is the re-weighting hyper-parameter, n is the total number of samples, y are the classes, γ = 1 is the focal loss hyper-parameter and pi = σ(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Balancing the network weights θ is performed via a MaxNorm constraint ∥θl,i∥2 2 ≤ δ2,∀i given in (Alshammari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2022) imposed on each class filter i in the last layer l of the network where δ is the L2-norm ball radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' We will reference a LCB-based optimisation where weight balancing is performed via LWB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Methodologically, this described architecture ap- proaches the learning of behavioural great ape actions via five key capabilities: 1) utilisation of multiple rel- evant input modalities across an entire video snippet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 2) effective streamed content encoding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 3) fusion into a single embedding space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 4) metric space optimisa- tion so that distances naturally reflect semantic sim- ilarity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' and 5) taking into account class imbalances common to the domain content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 5 EXPERIMENTS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='1 General Training Setup We train our architecture via SGD optimisation using batch size 32 and learning rate 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Feature extrac- tor backbones are initialised with Kinetics-400 (Kay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2017) pre-trained weights and training runs are distributed over 8 Tesla V100 GPUs for 100 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='2 Baselines and Stream Ablations As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 1, we first establish performance benchmarks for one and two stream baseline archi- tectures of our system (rows 2–5) against the cur- rent state-of-the-art (row 1), which uses a ResNet-18 backbone with focal loss LFL, SGD, and LSTM-based frame fusion (Sakib and Burghardt, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' As ex- pected, we confirmed that - using identical setups and losses - adding an optical flow stream is beneficial in the great ape domain mirroring HAR results (see rows 2 vs 4, and 3 vs 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Additionally, models trained using LRC consistently outperformed standard triplet loss LRC scenarios (see rows 2 vs 3, and 4 vs 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Fi- nally, a dual-stream version of our proposed architec- ture trained with LRC outperforms the state-of-the-art by a small margin (see rows 1 vs 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='3 Triple-Stream Recognition As given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 1 rows 6–8, our proposed triple- stream architecture significantly outperforms all base- lines with regards to top-1 accuracy, achieving up to 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='86%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Thus, explicit DensePose-C information ap- pears a useful information source for boosting be- havioural action recognition in great apes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' However, Table 1: Behavioural Action Recognition Benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Top-1 and average per-class (C-Avg) accuracy performance on the PanAf-500 dataset for the current state-of-the- art (row 1), single and dual-stream baselines (rows 2–5), and our triple-stream networks (rows 6–8) for different fu- sion methodologies and losses tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Models/Streams Fusion Loss Top-1 C-Avg Sakib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 2020 1 RGB+OF LSTM LFL 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='52% 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='33% Up to Dual-Stream 2 RGB only None LTL 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='50% 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='67% 3 RGB only None LRC 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='24% 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='76% 4 RGB+OF Avg LTL 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='90% 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='10% 5 RGB+OF Avg LRC 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='02% 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='97% Triple-Stream (Ours) 6 RGB+OF+DP Avg LRC 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='71% 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='61% 7 RGB+OF+DP Conv LRC 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='04% 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='31% 8 RGB+OF+DP Elem LRC 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='86% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='50% without LTR techniques all our triple-stream models are significantly outperformed by a dual-stream set- ting (row 5) with regards to average per-class accu- racy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' This reduction is caused by significantly poorer performance on minority classes (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Since the learned behavioural action embeddings are constructed as metric from the outset, they can be visualised meaningfully – we note that such data- driven visualisations are novel in the primatology do- main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 5 depicts such learned spaces for our data and architecture where, independent of stream cardi- nality, embeddings cluster the training data cleanly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' This is of course expected given above 99% top-1 training accuracy in all settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Yet, behavioural ac- tions of great apes are highly intricate as well as vari- able and, even with approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 144,000 training frames used, the model clearly shows signs of overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' As a result, test set embeddings exhibit significant cluster overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Sample groups representing sitting, standing, and walking, for instance, blend into one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' In addition to overfitting, this also highlights the transi- tional nature of these often temporarily adjacent and smoothly changing actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Thus, future temporally transitional ground truth labelling may be needed to represent behavioural great ape action in the PanAf- 500 dataset more authentically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='4 Fusing Streams When looking at the impact of information fusion methods on performance in more detail, we find that benchmarks vary significantly (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 1 rows 6–8) when we test averaging, element-wise multiplication, and convolutional fusion, as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Re- sults show that convolution and element-wise mul- tiplication improve performance slightly across both metrics when compared with averaging: top-1 accu- camera interaction climbing up climbing down hanging running sitting sitting on back standing walking Behavioural actions Single Stream (RGB) Kinetics pretrained (no training) Training Training Test Dual Stream (RGB+OF) Triple Stream (AllThree) Figure 5: Visualisations of Great Ape Behavioural Action Spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' A 2D t-SNE (Wattenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=', 2016) visualisation of the 128-dimensional training (top-right) and test (bottom-right) embeddings produced by the single, dual and three-stream network with convolutional fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' We can see that training set embeddings from all classes are clustered cleanly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' In contrast, test set embeddings show significant overlap and only embeddings from majority classes form distinct clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' This is consistent with the high top-1 accuracy and relatively low average per-class accuracy reported in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 1 racy improves by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='33% and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='1%, respectively (see rows 6–8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' However, the most significant gains are observed with respect to average per class accuracy which increases by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='44% for element-wise multipli- cation and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='7% for convolutional fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Learnable parameters in the convolution method clearly help blending information even when only fewer samples are available for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Building on this improve- ment, we will next investigate the impact of LTR methods in order to benefit tail class performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='5 Long-tail Recognition When grouping behavioural actions into head (cov- ering sitting, standing, and walking) and remain- ing tail classes based on frequency in the data (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 2), a significant performance gap becomes appar- ent even when using the so far best C-Avg performing model (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 2 row 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Employing LTR techniques can, however, reduce this gap and improve average per-class accuracy further as quantified across rows 2–4 in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 6 shows t-SNE visualisations of the three LTR triple-stream approaches when trained with convolutional feature fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Particularly for the class-balanced approaches and weight-balancing se- tups (two rightmost), tail class clusters appear more clearly separated and class overlap is generally re- duced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Thus, for the great ape domain underrepre- sented classes are indeed an effective source of infor- mation for improving action separability in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' 6 CONCLUSION In this work we introduced the first deep metric learn- ing system for great ape behavioural action recogni- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' We demonstrated that the proposed triple-stream architecture can provide leading state-of-the-art per- formance when tested on the PanAf-500 camera trap dataset covering 180,000 annotated frames across 500 videos taken in the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' We demonstrated that the ad- dition of a DensePose-C chimpanzee pose estimation stream into the embedding architecture is highly ef- fective and leads to system performance of 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='86% top-1 accuracy on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' We also showed that adding LTR techniques that address poor tail class performance to the system can improve the average per-class accuracy to 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='66% on the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Despite these improvements we note that both larger anno- tated datasets to counteract overfitting as well as more temporally blended forms of annotation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' action transition annotations) would benefit the authenticity of data-driven great ape behavioural representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' We hope that the research presented here sparks fur- ther interest in this vital application area for the bene- fit of endangered species such as great apes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We thank the Pan African Programme: ‘The Cultured Chimpanzee’ team and its collaborators for allowing the use of their data for this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' We thank Amelie Pettrich, An- tonio Buzharevski, Eva Martinez Garcia, Ivana Kirchmair, climbing up hanging running sitting sitting on back standing walking Logit adjustment No LTR augmentation Weight balanced CB (+focal loss) climbing down camera interaction Test Figure 6: Long-tail Test Embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' A 2D t-SNE visualisation of the 128-dimensional test embeddings produced by the three-stream network with convolutional fusion alone (leftmost) and augmented with each LTR technique;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' (i) logit adjustment (ii) CB (+focal loss) and (iii) weight balancing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' All LTR-augmented methods improve clustering of embeddings belonging to tail classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' They appear more clearly separated and exhibit less overlap when compared with the non-LTR method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Sebastian Sch¨utte, Linda Gerlach and Fabina Haas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' We also thank management and support staff across all sites;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' specif- ically Yasmin Moebius, Geoffrey Muhanguzi, Martha Rob- bins, Henk Eshuis, Sergio Marrocoli and John Hart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Thanks to the team at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='chimpandsee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='org particularly Briana Harder, Anja Landsmann, Laura K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Lynn, Zuzana Mach´aˇckov´a, Heidi Pfund, Kristeena Sigler and Jane Wid- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' The work that allowed for the collection of the dataset was funded by the Max Planck Society, Max Planck Society Innovation Fund, and Heinz L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Krekeler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' In this respect we would like to thank: Ministre des Eaux et Forˆets, Minist`ere de l’Enseignement sup´erieur et de la Recherche scientifique in Cˆote d’Ivoire;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Institut Congolais pour la Conservation de la Nature, Minist`ere de la Recherche Scientifique in Demo- cratic Republic of Congo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Forestry Development Authority in Liberia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Direction Des Eaux Et Forˆets, Chasses Et Con- servation Des Sols in Senegal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Makerere University Biolog- ical Field Station, Uganda National Council for Science and Technology, Uganda Wildlife Authority, National Forestry Authority in Uganda;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' National Institute for Forestry De- velopment and Protected Area Management, Ministry of Agriculture and Forests, Ministry of Fisheries and Environ- ment in Equatorial Guinea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' This work was supported by the UKRI CDT in Interactive AI under grant EP/S022937/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Table 2: LTR-enabled Behavioural Action Recogni- tion Benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Average per-class accuracy for our triple-stream network with convolutional fusion for best performing non-LTR method (row1), and three LTR ap- proaches (rows 2–4) targetting poor tail class performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content=' Method/Loss C-Avg Head Tail Non-LTR Triple-Stream 1 LRC 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='31 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='57 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='78 LTR Triple-Stream 2 LLA 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='76 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='22 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='7 3 LCB 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E0T4oBgHgl3EQfxAH-/content/2301.02642v1.pdf'} +page_content='56 77.' 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method. The dataset used contains 19 billion events from +the PHENIX experiment’s Run 14 Au + Au dataset at √sNN = 200 GeV. +PHENIX has measured a J/ψ v2 in a centrality range of 10 − 60% that is +consistent with zero. Taken together with results from LHC the measure- +ment of v2, which is consistent with zero may indicate that J/ψ production +by coalescence is not significant at forward rapidity at RHIC energy. +1. Introduction +The QGP has been found to exhibit a nearly perfect fluid behavior [1]. +This behavior manifests itself as strong correlations between particles pro- +duced in nuclear collisions. Presently, the detailed interactions of the heavy +quarks in the QGP medium are under investigation and, because heavy fla- +vor quarks will have relatively larger masses, they may not be thermalized +and flow with the medium. +The production of J/ψ in p+p collisions is +theoretically well understood because they are produced in hard scattering +processes. This feature in addition to their production in hard scattering +events in the initial stages of the collision make them ideal probes for testing +the properties of the QGP medium. However, in nucleus+nucleus collisions +some of the produced J/ψ mesons may be dissolved by the QGP, which may +create anisotropies in the observed J/ψ azimuthal distributions due to the +different path length in the medium. Additionally, a similar signal may be +created if the J/ψ thermalizes inside the medium and follows the pressure +gradients as lighter particles do, or the J/ψ may dissociate, and the charm +∗ Presented at the 29th International Conference on Ultrarelativistic Nucleus-Nucleus +Collisions (Quark Matter 2022) +(1) +arXiv:2301.04186v1 [nucl-ex] 10 Jan 2023 + +2 +QM˙Proceedings˙Bichon +printed on January 12, 2023 +quarks could equilibrate which could lead to J/ψ regeneration. We present +a preliminary result for J/ψ v2 using the PHENIX Run14 Au+Au dataset +at √sNN = 200 GeV. +2. Data Analysis & Methodology +2.1. Dataset and Detectors +In this analysis, we use the Run 14 Au+Au Muon Arm dataset at +√sNN = 200 GeV containing 19 billion events. The dimuon decay channel +is used to reconstruct candidate J/ψ mesons. The PHENIX experiment has +a unique coverage at forward rapidity with muon identification. This in ad- +dition to the large dataset of Au+Au collisions collected in 2014 allows for +a statistically improved measurement of J/ψ elliptic flow at RHIC energies. +The key detector in this analysis is the Forward Silicon Vertex Detector +(FVTX). With the FVTX, an increase in precision vertexing capabilities +was added to the muon spectrometers, enabling the rejection of muons from +the decay of relatively long-lived particles, the rejection of muons from the +decays of relatively long-lived particles, and an additional way of determin- +ing the event plane [2]. +2.2. Combinatorial Background Subtraction +To obtain a pure signal for the J/ψ from dimuon mass distributions we +employ event-mixing as the standard method of removing the background +dimuons. +For this event-mixing method, the background is constructed +from dimuon pairs of opposite sign, but the single muons come from differ- +ent events. Mixed event dimuon pairs are only formed if two events have a +centrality closer than 5%, a Z vertex closer than 0.75 cm and a event plane +angle closer than π/20 rad. +Using events instead of individual dimuons +allows us to increase the likelihood that we are using combinatorial back- +ground dimuons. A normalization factor must be applied for the background +which can be obtained by using the ratio of like-sign pairs from the same +event to like-sign pairs from mixed events. The signal is then obtained by +the subtraction of the normalized background from the foreground. +2.3. Fitting the Dimuon Mass Distribution +In the fitting of the mass distributions, we assume the shape of the +J/ψ signal to be a Crystal Ball function, and given the statistical precision +of the dataset, we also apply the same shape to the Ψ(2S) to avoid their +inclusion in the higher mass J/ψ region. The parameters of the Crystal Ball +function are obtained using J/ψ embedded Monte Carlo simulation data. +We produce simulated mass distributions for low/high pT and South/North + +QM˙Proceedings˙Bichon +printed on January 12, 2023 +3 +arm rapidities, fitting the distributions allowing for the function to have +free (α, n, ¯x, and σ) parameters. The J/ψ count for each distribution is +obtained by the integral of the J/ψ crystal ball function in the fit (see Figure +1). +Fig. 1. Mass distributions using mixed-event subtraction for the unweighted “stan- +dard” set. These are binned by pT in each column, and rapidity+∆φ angle for +each row. The green/dashed curve is a Crystal Ball fitted to the J/ψ peak, the +blue/dashed-dot curve is a Crystal Ball fitted to the ψ(2S) peak, the red/dotted +curve is an exponential fitted to the remaining background after subtraction, and +the black/solid curve is the total fit. + +0 +Nj/±= 1422±70 +Nj/ =1683±74 +150 +South Arm - 0 +100 +80 +PH米ENIX +PH米ENIX +40 +100 +PH米ENIX +PH米ENIX +60 +preliminary +preliminary +preliminary +preliminary +50 +40 +20+ +20 +2 +2.5 +3 +3.5 +4.5 +2 +2.5 +3 +3.5 +4.5 +2 +2.5 +3 +3.5 +4.5 +2.5 +3 +3.5 +4.5 +Mass [GeV/c2] +Mass [GeV/c2 +Mass [GeV/c2] +Mass [GeV/c2] +300日 +250 +80 +π-2 +T+T +x= 3.125 ±0.003 GeV/c2 +250E + = 3.125 ± 0.003 GeV/c2 +x=3.118 ±0.004 GeV/c2 +x = 3.118 ± 0.004 GeV/c2 +VI + = 0.135 ± 0.003 GeV/c2 + = 0.135 ± 0.003 GeV/c2 + = 0.160 ± 0.002 GeV/c2 + = 0.160 ± 0.002 GeV/c2 +200 +α= 1.19 +α= 1.19 +α = 1.00 +α = 1.00 +v +n = 3.94 +200 +n = 3.94 +n = 4.06 +60 +100 +n = 4.06 +150 +V +Nj/ = 1374±66 +150E +Nj/Φ =1582±72 +N +π一4 +100 +40 +PH米ENIX +100E +PH米ENIX +PH米ENIX +PH米ENIX +South Arm - +preliminary +preliminary +preliminary +preliminary +50 +20 +-50 +50田 +50 +2 +2.5 +3.5 +4.5 +2.5 +4.5 +2.5 +4.5 +2.5 +Mass [GeV/c2] +Mass [GeV/c2] +Mass [GeV/c2] +Mass [GeV/c2] +140F +80 +允-4 +120 +3.147±0.006GeV/c2 +120 +x=3.147±0.006 GeV/c2 +x = 3.159 ± 0.006 GeV/c2 +50E +x = 3.159 ± 0.006 GeV/c2 +VI + = 0.160 ± 0.002 GeV/c2 + = 0.160 ± 0.002 GeV/c2 + = 0.146 ± 0.006 GeV/c2 + = 0.146 ± 0.006 GeV/c2 +>- +100 +α = 0.63 +100 +α = 0.63 +60 +α= 1.17 +α = 1.17 +n = 7.60 +n= 7.60 +n = 2.55 +40 +n = 2.55 +80E +Nj/ =902±63 +[Nj/=1013±63 +30 +60 +60 +PH*ENIX +North Arm - +PH米ENIX +PH米ENIX +PH米ENIX +20 +preliminary +40 +preliminary +preliminary +preliminary +20 +10吨 +-20 +2 +2.5 +3.5 +2.5 +2.5 +3.5 +4.5 +2.5 +Mass [GeV/c2] +Mass [GeV/c2 +Mass [GeV/c2] +Mass [GeV/c2] +160 +150 +100 +40 +π-2 +X= 3.147 ± 0.006 GeV/c2 +x=3.147 ±0.006 GeV/c2 +x=3.159±0.006 GeV/c2 +x= 3.159 ± 0.006 GeV/c2 +VI +140 + = 0.160 ± 0.002 GeV/c2 + = 0.160 ± 0.002 GeV/c2 + = 0.146 ± 0.006 GeV/c2 +g = 0.146 ± 0.006 GeV/c2 +α = 0.63 +α = 0.63 +80 +α = 1.17 +30 +α = 1.17 +120 +n = 7.60 +100 +n = 7.60 +n = 2.55 +n = 2.55 +> +100 +60 +"j/ = 893±60 +Nj/=474±35 +π-4 +20 +PH米ENIX +50 +PH米ENIX +PH米ENIX +PH米ENIX +North Arm - +60E +preliminary +preliminary +preliminary +10- +preliminary +40 +20 +-20E +50 +20— +2 +2.5 +3 +3.5 +4.5 +2.5 +3 +3.5 +4.5 +2 +2.5 +3 +3.5 +4.5 +2.5 +3 +3.5 +Mass [GeV/c2] +Mass [GeV/c2] +Mass [GeV/c2]4 +QM˙Proceedings˙Bichon +printed on January 12, 2023 +2.4. Event Plane Method and Measuring v2 +We are primarily using the In/Out ratio method, which is an event plane +method [3] that uses the counts of the J/ψ in bins of ∆φ to measure v2. +The In/Out ratio method splits the distributions into 2 bins of ∆φ one in +plane with the event plane and the other out of plane. We measure v2 using +this method by looking at the difference between these bins. If there is no +preference in either plane, we would observe a flow around zero. +2.5. Systematic Uncertainties +The systematic uncertainties are determined by changing various aspects +of the analysis. As of this time, we have employed changing the primary +detector of the analysis from the FVTX to the Central Arm Spectrometers +(CNT), which covers a different pseudorapidity range. +We have used a +different method for our combinatorial background subtraction, the like-sign +method, which constructs the background with dimuon pairs of the same +sign (µ+µ+ and µ−µ−) that come from the same event. The uncertainty in +the normalization factor in the event-mixing method was also incorporated +into the systematic uncertainty. The last systematic uncertainty we consider +comes from the mass fitting of the dimuon distribution, where the shape +of the continuum distribution was assumed to be an exponential function, +and the uncertainty in this assumption can be explored by assuming no +continuum contribution in the J/ψ mass region. +3. Results +Figure 2 shows the pT -dependent J/ψ v2. +The measurement in this +analysis for PHENIX Run 14 at forward rapidity in a centrality range of 10 +- 60% is shown in red. The measurement made by STAR at mid-rapidity +and in a centrality range of 10-40% is shown in black. The ALICE result +at forward rapidity in a centrality range of 20-40% is shown in blue. Boxes +surrounding the data points represent systematic uncertainties. +PHENIX observes a larger suppression of J/ψ yield in forward rapidity +when compared to mid-rapidity. This is contrary to expectations, because +effects that dissolve the J/ψ have been determined to be stronger at mid- +rapidity [4]. To understand this observation we begin by looking into the +production of c¯c pairs. The majority of c¯c pairs per event in central collisions +at RHIC are produced at mid-rapidity. At LHC energies, less suppression +is observed, where many more c¯c pairs per event in central collisions are +produced [5]. To explain this behavior, theoretical models require a contri- +bution of coalescence via a recombination mechanism between charm and +anticharm quarks [6]. It was found that the strength of this coalescence + +QM˙Proceedings˙Bichon +printed on January 12, 2023 +5 +effect increases with the initial number of produced c¯c pairs relative to the +total number of quarks, increasing with the collisions energy. +At LHC energies, a nonzero v2 is observed, this is in line with J/ψ +formed by coalescence in the QGP medium, and carrying the azimuthal +anisotropy of the system [7]. At RHIC energies, STAR has measured v2 that +is consistent with zero, but due to limited statistics remains inconclusive [8]. +With coalescence being the dominant mechanism for nonzero J/ψ v2 it +should follow that systems where fewer c¯c pairs are formed should have a +smaller azimuthal anisotropy. +Fig. 2. Plot of pT dependent J/ψ v2. The PHENIX result in light gray/red/circle +is compared to STAR [8] in black/star and ALICE [7] gray/blue/square. +From the figure we can see the clear nonzero v2 measured by ALICE. +Although the ALICE measurement is at a much higher energy, we know + +0.3 +Au+Au → J/ + X /Snn = 200 GeV +PHENIX Run14. 10 - 60%. 1.2 < Iyl < 2.2 +★ STAR, 10 - 40%, lyl < 1 (PRL 111, 052301 (2013)) +0.2 +Pb+Pb → J/Φ + X /Snn = 5.02 TeV +ALICE, 20 - 40%, 2.5 < lyl < 4.4 (JHEP 10 (2020) 141) +0.1 +-0.1 +PH米ENIX +-0.2 +preliminary +-0.3 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +pT [GeV/c]6 +QM˙Proceedings˙Bichon +printed on January 12, 2023 +v2 does not scale with energy for J/ψ, so it makes for a good comparison +that the ALICE result which is clearly nonzero is different from our mea- +surement. In our measurement, we see a v2 that is clearly consistent with +zero across all pT bins. The systematic uncertainties were conservatively +estimated, not taking into account cancellations or correlations of uncer- +tainties from different sources. Additional data from Run 16 of RHIC will +be included in the final results, and we expect that both statistical and +systematic uncertainties will be significantly reduced. +4. Conclusion and Outlook +We have presented PHENIX Run 14 pT -dependent J/ψ v2 at forward +rapidity at √sNN = 200 GeV. PHENIX has measured a J/ψ v2 that is +consistent with zero. We have determined that the ALICE result, where +there is clearly nonzero v2, is distinctly different from our measurement, +and that forward and mid-rapidity results at RHIC are consistent, but the +uncertainties are still large. In the future, we will incorporate Run 16 data +in our measurement, essentially doubling the current dataset and reducing +statistical uncertainties accordingly. We also plan to study open heavy flavor +v2 to obtain a more complete understanding of the heavy flavor dynamics +at RHIC. +REFERENCES +[1] Ulrich Heinz. +The strongly coupled quark–gluon plasma created at RHIC. +Journal of Physics A: Mathematical and Theoretical, 42(21):214003, May 2009. +[2] C. Aidala et al. The PHENIX forward silicon vertex detector. Nuclear Instru- +ments and Methods in Physics Research Section A: Accelerators, Spectrometers, +Detectors and Associated Equipment, 755:44–61, Aug 2014. +[3] A. M. Poskanzer and S. A. Voloshin. Methods for analyzing anisotropic flow in +relativistic nuclear collisions. Physical Review C, 58(3):1671–1678, Sep 1998. +[4] A. Adare et al. J/ψ suppression at forward rapidity in Au+Au collisions at +√sNN = 200 GeV. Physical Review C, 84:054912, Nov 2011. +[5] Anton Andronic, Peter Braun-Munzinger, Krzysztof Redlich, and Johanna +Stachel. Decoding the phase structure of QCD via particle production at high +energy. Nature, 561(7723):321–330, Sep 2018. +[6] H. Pereira Da Costa et al. Charmonium production in Pb–Pb collisions with +ALICE at the LHC. Nuclear Physics A, 956:705–708, Dec 2016. +[7] S. Acharya et al. J/ψ elliptic and triangular flow in Pb-Pb collisions at √sNN += 5.02 TeV. Journal of High Energy Physics, 2020(10), Oct 2020. +[8] L. Adamczyk et al. +Measurement of J/ψ Azimuthal Anisotropy in Au+Au +Collisions at √sNN = 200 GeV. Physical Review Letters, 111(5), Aug 2013. + diff --git a/1NE2T4oBgHgl3EQf5Ahd/content/tmp_files/load_file.txt b/1NE2T4oBgHgl3EQf5Ahd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..91d9170553a3d33b61925b8e40801d3b27309b52 --- /dev/null +++ b/1NE2T4oBgHgl3EQf5Ahd/content/tmp_files/load_file.txt @@ -0,0 +1,277 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf,len=276 +page_content='Elliptic flow measurement of J/ψ in PHENIX Run14 Au+Au at √sNN = 200 GeV ∗ Luis Bichon III (for the PHENIX collaboration, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content='7430208) Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235 USA We obtain the first measurement of J/ψ elliptic flow at RHIC energies in forward rapidity using data from the PHENIX detector and applying an event plane method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' The dataset used contains 19 billion events from the PHENIX experiment’s Run 14 Au + Au dataset at √sNN = 200 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' PHENIX has measured a J/ψ v2 in a centrality range of 10 − 60% that is consistent with zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' Taken together with results from LHC the measure- ment of v2, which is consistent with zero may indicate that J/ψ production by coalescence is not significant at forward rapidity at RHIC energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' Introduction The QGP has been found to exhibit a nearly perfect fluid behavior [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' This behavior manifests itself as strong correlations between particles pro- duced in nuclear collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' Presently, the detailed interactions of the heavy quarks in the QGP medium are under investigation and, because heavy fla- vor quarks will have relatively larger masses, they may not be thermalized and flow with the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' The production of J/ψ in p+p collisions is theoretically well understood because they are produced in hard scattering processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' This feature in addition to their production in hard scattering events in the initial stages of the collision make them ideal probes for testing the properties of the QGP medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' However, in nucleus+nucleus collisions some of the produced J/ψ mesons may be dissolved by the QGP, which may create anisotropies in the observed J/ψ azimuthal distributions due to the different path length in the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' Additionally, a similar signal may be created if the J/ψ thermalizes inside the medium and follows the pressure gradients as lighter particles do, or the J/ψ may dissociate, and the charm ∗ Presented at the 29th International Conference on Ultrarelativistic Nucleus-Nucleus Collisions (Quark Matter 2022) (1) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content='04186v1 [nucl-ex] 10 Jan 2023 2 QM˙Proceedings˙Bichon printed on January 12, 2023 quarks could equilibrate which could lead to J/ψ regeneration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' We present a preliminary result for J/ψ v2 using the PHENIX Run14 Au+Au dataset at √sNN = 200 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' Data Analysis & Methodology 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' Dataset and Detectors In this analysis, we use the Run 14 Au+Au Muon Arm dataset at √sNN = 200 GeV containing 19 billion events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' The dimuon decay channel is used to reconstruct candidate J/ψ mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' The PHENIX experiment has a unique coverage at forward rapidity with muon identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' This in ad- dition to the large dataset of Au+Au collisions collected in 2014 allows for a statistically improved measurement of J/ψ elliptic flow at RHIC energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' The key detector in this analysis is the Forward Silicon Vertex Detector (FVTX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' With the FVTX, an increase in precision vertexing capabilities was added to the muon spectrometers, enabling the rejection of muons from the decay of relatively long-lived particles, the rejection of muons from the decays of relatively long-lived particles, and an additional way of determin- ing the event plane [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' Combinatorial Background Subtraction To obtain a pure signal for the J/ψ from dimuon mass distributions we employ event-mixing as the standard method of removing the background dimuons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' For this event-mixing method, the background is constructed from dimuon pairs of opposite sign, but the single muons come from differ- ent events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' Mixed event dimuon pairs are only formed if two events have a centrality closer than 5%, a Z vertex closer than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content='75 cm and a event plane angle closer than π/20 rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' Using events instead of individual dimuons allows us to increase the likelihood that we are using combinatorial back- ground dimuons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' A normalization factor must be applied for the background which can be obtained by using the ratio of like-sign pairs from the same event to like-sign pairs from mixed events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' The signal is then obtained by the subtraction of the normalized background from the foreground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' Fitting the Dimuon Mass Distribution In the fitting of the mass distributions, we assume the shape of the J/ψ signal to be a Crystal Ball function, and given the statistical precision of the dataset, we also apply the same shape to the Ψ(2S) to avoid their inclusion in the higher mass J/ψ region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' The parameters of the Crystal Ball function are obtained using J/ψ embedded Monte Carlo simulation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' We produce simulated mass distributions for low/high pT and South/North QM˙Proceedings˙Bichon printed on January 12, 2023 3 arm rapidities, fitting the distributions allowing for the function to have free (α, n, ¯x, and σ) parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' The J/ψ count for each distribution is obtained by the integral of the J/ψ crystal ball function in the fit (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' Mass distributions using mixed-event subtraction for the unweighted “stan- dard” set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' These are binned by pT in each column, and rapidity+∆φ angle for each row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' The green/dashed curve is a Crystal Ball fitted to the J/ψ peak, the blue/dashed-dot curve is a Crystal Ball fitted to the ψ(2S) peak, the red/dotted curve is an exponential fitted to the remaining background after subtraction, and the black/solid curve is the total fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQf5Ahd/content/2301.04186v1.pdf'} +page_content=' 0 0.7 +and/or neuroretinal rim narrowing with repeatable glaucomatous VF defects and non- +occludable angles on gonioscopy whereas non-glaucomatous (normal) eyes were those with +an IOP < 21 mmHg and normal VF examinations. Subjects with corneal abnormalities that +potentially can reduce the quality of the OCT scans and with ONH disorders other than +glaucoma were excluded from the studies. +Based upon the mean deviation (MD) of the 24-2 or 30-2 VF, all glaucoma subjects +were further split into three glaucoma severity groups [21]: (1) mild glaucoma (MD ≥ -6.00 +dB); (2) moderate glaucoma (MD of -6.01 to -12.00 dB); and (3) advanced glaucoma (MD < - +12.00 dB). Even though this classification has its limitations [22], it remains a standard and +can be used as a good first indicator for staging functional damage. More information on the + + +7 +demographics of the four groups (i.e. normal, mild, moderate, and advanced) can be found +in Table 1. +Optical Coherence Tomography Imaging + +Each patient from both cohorts had their ONH imaged with the same spectral domain +OCT device (Spectralis, Heidelberg Engineering, Germany). All OCT scans (horizontal raster +scans) covered an area of 15x10 centered on the ONH and the number of B-scans varied +between 49 and 97 B-scans (distance between B-scans varied from approximately 35 to 70 +µm) with 384 A-scans per B-scan (approximately 11.5 µm between A-scans) and 496 pixels +per A-scan (axial resolution of 3.87 µm/pixel). Images were acquired using signal averaging, +eye tracking, and the enhanced depth imaging modality of the Spectralis OCT device. +Describing the Structural Phenotype of Glaucoma as a Function of Glaucoma +Severity + +In the following sections, we introduce two different approaches to study the complex +structural changes of the ONH as a function of glaucoma severity. In the first, we performed +a comprehensive 3D structural analysis of the ONH using ‘human-defined’ 3D structural +parameters of the ONH (total of 10 parameters) describing the morphologies of both neural +and connective tissues. In the second, we used a relatively recent geometric deep learning +method (i.e. PointNet) to discover important 3D structural features differentiating ONHs from +different glaucoma severity groups. An overview of both approaches is shown in Figure 1. +Approach 1 for Describing the Structural Phenotype of Glaucoma – ONH +Parameters + +For this approach, all ONH tissues were segmented in 3D (from the OCT scans), from +which all ONH structural parameters were automatically extracted. + + +8 +AI-based Segmentation of ONH Tissues. We automatically segmented all raw OCT +volume scans of the ONH using REFLECTIVITY (Reflectivity, Abyss Processing Pte Ltd, +Singapore) – a software that was developed from advances in AI-based ONH segmentation +[23] (see Figure 1a, b). More specifically, we automatically labelled the following ONH tissue +groups: (1) the retinal nerve fiber layer (RNFL) and the prelamina tissue (PLT); (2) the ganglion +cell inner plexiform layer (GCL+IPL); (3) all other retinal layers (ORL); (4) the retinal pigment +epithelium (RPE) with Bruch’s membrane (BM) and the BM opening (BMO) points; (5) the +choroid; (6) the OCT-visible part of the peripapillary sclera including the scleral flange; and (7) +the OCT-visible part of the LC. In almost all OCT volume scans, the posterior boundaries of the +sclera and LC were not visible and could therefore not be segmented. +Automated extraction of ONH parameters. Using the software REFLECTIVITY, we +extracted the following parameters: (1) the average RNFL thickness (RNFLT) in each octant +(i.e. temporal [T], superior-temporal [ST], superior [S], superior-nasal [SN], nasal [N], inferior- +nasal [IN], inferior [I], and inferior-temporal [IT]) calculated at a distance of 1.5 times the BMO +radius (BMOR) from the centre of BMO; (2) the average minimum rim width (MRW) in each +octant defined as the minimum distance from a BMO point to a point on the inner limiting +membrane (ILM); (3) the average ganglion cell complex thickness (GCCT) in each octant +evaluated at the same location than the RNFLT; (4) the average choroidal thickness (ChT) in +each octant at the same distance as that used for the RNFLT; (5) the prelamina depth (PLD) +defined as the distance from the BMO center to a point on the ILM (perpendicular to the BMO +plane); (6) the minimum prelamina thickness (MPT); (7) the LC depth (LCD) defined as the +distance from the BMO centre to a point on the anterior LC boundary (perpendicular to the +BMO plane); (8) the LC global shape index (LC-GSI) that summarizes the shape of the anterior +LC boundary into a single number [24]; (9) the peripapillary scleral angle (PPSA) representing + + +9 +the amount of scleral bowing and defined as the angle between two parallel lines to the +anterior scleral boundary in the nasal-temporal plane; and (10) the BMO area defined as the +area of the best-fit ellipse to the BMO points. A visualization of the extracted ONH parameters +is shown in Figure 1c. + +Statistical analysis. All parameters were compared across all 4 groups (normal, mild, +moderate, and advanced). All statistical analyses were performed using R (version 4.2.1) and +RStudio (version 2022.07.1 for macOS). ONH parameters that were extracted in each octant +were reported as mean  standard deviation and single valued ONH parameters were +presented as box plots. One-way ANOVA with post-hoc Tukey HSD test was used for the +comparisons. P value for significance was set at <0.05. +Approach 2 for Describing the Structural Phenotype of Glaucoma – PointNet + +PointNet, a deep neural network from the group of geometric deep learning +algorithms, can learn from complex 3D shapes, such as that of the ONH, if they are +represented as 3D point clouds. In contrast to our first approach, which relied on ‘human- +defined’ ONH parameters, PointNet allows us to identify important structural landmarks that +can differentiate ONHs from the four different glaucoma severity groups, without previous +inputs or guidance. + +Representation of the ONH structure as 3D point cloud. We described the structure +of a given ONH as a 3D point cloud which then was used as input to PointNet. To do so, we +first identified the anterior boundaries of all tissue layers in the segmented OCT scan. Each +anterior boundary voxel was then represented as a 3D point (see Figure 1d). The final point +cloud consisted of about 20,000 points for each ONH (see Figure 1e). Additionally, for each +point, we extracted the local tissue thickness (minimum distance between anterior and +posterior boundary). In summary, we assigned four values to every point: its position in the + + +10 +3D space ([x, y, z]-coordinate) and its local tissue thickness (not applicable for the sclera and +LC). To homogenize the data across all ONHs, the centre of BMO was set as origin of the +coordinate system [x=0, y=0, z=0] and the normal of BMO plane (best-fit plane to the BMO +points) was aligned with the axial direction of the scan. The more interested reader is referred +to our previous publication on geometric deep learning for glaucoma diagnosis [25]. + +Glaucoma severity classification. PointNet was specifically designed to process and +learn from 3D point clouds such as the one shown in Figure 1. We used the same architecture +as in the original publication [18], except that we implemented a max pooling layer of +dimension 256. To identify important 3D structural features of the ONH at different stages of +glaucoma, we trained three PointNet classification networks to differentiate between: (1) +normal and mild glaucoma subjects (normal-mild); (2) mild and moderate glaucoma subjects +(mild-moderate); and (3) moderate and advanced glaucoma subjects (moderate-advanced). +To assess the performance of the three binary classification networks, we split each +respective dataset (i.e. normal-mild, mild-moderate, and moderate-advanced) in training +(70%), validation (15%), and test (15%) sets. To improve performance and reduce overfitting, +we used data augmentation techniques such as random cropping, random rotations, random +rigid translations, random sampling (i.e. randomly picking a subset of points from the input +point cloud), oversampling to reduce data imbalance, and additive Gaussian noise where +applicable. A five-fold cross validation study was performed (using the train and validation +set) to tune hyperparameters and we reported the area under the receiver operating +characteristic curves (AUCs) of the model with the best performing hyperparameters as mean +± standard deviation. All models were trained on a Nvidia RTX A5000 GPU card until optimum +performance was reached in the validation set. + + +11 +Identification of important 3D structural features of the ONH. The specific +architecture of PointNet inherently allowed us to identify regions of the ONH important for +the differentiation of different glaucoma severity groups by extracting all points that +contributed to the final classification score – the so-called critical points. For each +classification group (i.e. normal-mild, mild-moderate, and moderate-advanced), we extracted +critical points from all ONHs of the respective test set (networks trained on respective training +set using tuned hyperparameters). Comparing the locations of these points between the +three groups allowed us to draw conclusion on the characteristic 3D structural changes of the +ONH at different stages of glaucoma. +Visualization of critical points. To better visualize the location of the resulting critical +points, we first constructed an average ONH geometry (represented by the average anterior +boundaries of each segmented tissue) for each of the three classification groups, i.e. normal- +mild, mild-moderate, and moderate-advanced. For each group, we then projected the critical +points (closest point projection) onto their corresponding anterior tissue boundary of the +respective average ONH geometry and visualized them as 3D point cloud density maps. A +density measure for each point was obtained by counting the neighbouring points within a +75 μm radius sphere. Since all critical points were projected on an average ONH geometry, +such a density map should highlight landmarks of the ONH that exhibit distinct 3D structural +changes between the different stages of glaucoma (represented as a cluster of red points in +the point cloud density maps). + + + + +12 +Results +Approach 1 – Statistical Analysis of ONH Parameters +We observed that the majority of ONH structural changes occurred in the early +glaucoma stage (normal to mild). These changes were also the most substantial in terms of +their size or magnitude. Specifically, we noted a decrease in average RNFLT (average over all +sectors) from 112  26 µm to 83  29 µm (Figure 2a), a decrease in average MRW from 256  +60 µm to 169  55 µm (Figure 2b), a decrease in average GCCT from 154  26 µm to 124  30 +µm (Figure 2c), no change in average ChT (Figure 2d), an increase in PLD from 136  195 µm +to 288  199 µm (Figure 2e), a decrease in MPT from 146  116 µm to 63  70 µm (Figure 2f), +an increase in LCD from 410  109 µm to 468  132 µm (Figure 2g), a decrease in LC-GSI from +-0.37  0.42 to -0.61  0.33 (Figure 2h), an increase in PPSA from 5.4  4.6 degree to 9.5  6.2 +degree (Figure 2i), and an increase in BMOA from 2.15  0.5 mm2 to 2.28  0.5 mm2 (Figure +2j). +Following substantial structural changes of the ONH in the early stage of glaucoma, +most ONH parameters showed a plateau effect, with little change from mild to moderate +glaucoma. Only RNFLT (average), GCCT (average), and MRW (average) showed a significant +decrease from 83  29 to 71  30 µm, 124  30 to 111  32 µm, and 169  55 to 159  56 µm, +respectively. +In the later stages of glaucoma (moderate to advanced), we observed significant +structural changes of the ONH, but they were much less pronounced in terms of their +magnitude compared to those seen in the early stages. In detail, the average RNFLT decreased +from 71  30 µm to 50  25 µm (Figure 2a), the average MRW decreased from 159  56 µm +to 126  46 µm (Figure 2b), the average GCCT decreased from 111  32 µm to 88  27 µm + + +13 +(Figure 2c), the LCD increased from 459  121 to 502  147 µm (Figure 2g), and the BMOA +decreased from 2.30  0.58 mm2 to 2.12  0.42 mm2 (Figure 2j). The ChT (Figure 2d), the PLD +(Figure 2e), the MPT (Figure 2f), the LC-GSI (Figure 2h), and the PPSA (Figure 2i) showed no +significant change. +If we were to examine regional variations, we noted that structural changes of the +RNFLT, MRW, and GCCT were more pronounced (higher in magnitude) in both the superior +and inferior octants of the ONH. This was true throughout all stages of glaucoma. In these +sectors, we observed that the decrease in MRW slowed as glaucoma severity increased. +Specifically, in the early stage of glaucoma (normal to mild), MRW decreased in the superior +octant from 295  64 µm to 192  58 µm while in the later stage (moderate to advanced), the +decrease was smaller from 179  58 µm to 133  49 µm (Figure 2b). In contrast, RNFLT and +GCCT decreased linearly as glaucoma severity increased. In the early stage of glaucoma +(normal to mild), RNFLT and GCCT in the superior octant decreased from 163  31 to 122  +34 µm and 200  31 to 160  34 µm, respectively, while in the later stage (moderate to +advanced), the decrease was from 102  35 to 61  31 µm and 141  35 to 99  32 µm (Figure +2a, 2c). With the exception of the inferior octant of the ONH, we did not observe any +significant changes in the ChT with glaucoma severity (Figure 2d). +Approach 2 – Performance Assessment + +Using PointNet, we were able to differentiate ONHs from different glaucoma severity +groups. The normal-mild glaucoma classification showed the best performance (AUC: 0.94  +0.02), followed by the moderate-advanced (AUC: 0.80  0.04) and mild-moderate glaucoma +classification (AUC: 0.68  0.08). + + +14 +Approach 2 – Changes of Important 3D Structural Features of the ONH with +Glaucoma Severity +For each classification task (i.e. normal-mild, mild-moderate, and moderate- +advanced), we pooled all critical points from all ONHs (test set), mapped them onto the +corresponding average ONH geometry, and displayed them as a 3D point cloud density map +for all ONH tissues (Figure 3), or separately for each ONH tissue (Figure 4). +In general, we observed that critical points were present in both, neural (normal-mild: +57%, mild-moderate: 39%, moderate-advanced: 53%) and connective tissues (normal-mild: +43%, mild-moderate: 61%, moderate-advanced: 47%). More specifically, most of the critical +points were located in the RNFL+PLT (normal-mild: 53%, mild-moderate: 37%, moderate- +advanced: 47%), the sclera (normal-mild: 17%, mild-moderate: 15%, moderate-advanced: +11%), and the LC (normal-mild: 23%, mild-moderate: 43%, moderate-advanced: 31%). In +contrast, we observed almost no critical points in the other tissue layers, such as the GCC+IPL, +ORL, RPE, Choroid. +On a tissue level, we found that the critical points from the RNFL of all three +classification tasks formed an hourglass pattern with points mainly located in the superior +and inferior quadrant. In addition, in the normal-mild glaucoma classification, critical points +from the RNFL were mostly located around the neuro-retinal rim whereas in the moderate- +advanced glaucoma classification, these points moved more outwards to the peripheral +region of the ONH. Interestingly, we also found that in the normal-mild and mild-moderate +classification most of the critical points from the LC were located near the LC insertion zone +in the superior (normal-mild) and superior + inferior quadrant (mild-moderate) whereas in + + +15 +the moderate-advanced classification, critical points were more spread out over the entire +LC. + +Discussion + +In this study, we were able to describe the 3D structural phenotype of glaucoma as a +function of severity using two separate approaches. In the first, we extracted ‘human-defined’ +3D structural parameters of the ONH and compared them across four different groups: +normal, mild, moderate, and advanced. In the second, we represented the complex structure +of the ONH as a 3D point cloud and used PointNet to uncover the structural landmarks that +were the most affected by glaucoma severity without any human input. Overall, we found +that the structural features of both neural and connective tissues contributed to the +structural phenotype of glaucoma, and that each of our proposed method could provide its +own unique knowledge. + +In this study, we found that after substantial structural changes of the ONH in the early +stage of glaucoma (normal to mild), almost all ONH parameters reached a plateau, with less +change in the later stages (mild to moderate and moderate to advanced). This is in good +agreement with previous studies that investigated the structure-function relationship and +reported a considerable structural loss before any functional VF defects were detectable [26- +28]. Some of these studies suggested a “tipping point” in the early stage of glaucoma (at about +– 3 dB MD) from which onwards even small structural changes were associated with a +relatively large decrease in MD value [26, 28]. One should also keep in mind that MD values +are usually reported on a logarithmic scale (non-linear scale). For instance, a shift in MD value +from 0 to -6 dB would imply a much larger loss in visual sensitivity compared to a shift from - + + +16 +6 to -12 dB on a linear scale [29]. Therefore, the observed plateau effect might be a result of +reporting MD values on a logarithmic scale. However, further research is needed to verify +such a hypothesis. + +Furthermore, we found that critical points were present in both neural (normal-mild: +57%, mild-moderate: 39%, moderate-advanced: 53%) and connective tissues (normal-mild: +43%, mild-moderate: 61%, moderate-advanced: 47%) at all stages of glaucoma indicating that +the structural changes caused by glaucoma affected both types of tissue in the ONH. Our +findings are in line with previous research that suggested that the pathophysiology of +glaucoma is complex and cannot purely be characterized as a damage to the neural tissue in +the ONH (i.e. retinal ganglion cells) [11-13]. Despite these recent findings, current glaucoma +tests focus on assessing neural tissue health, ignoring any glaucomatous structural changes +of connective tissue in the ONH. In the future, the development of more comprehensive tests +that consider structural changes in both, neural and connective tissues, could potentially +improve the diagnosis and prognosis of glaucoma. +Additionally, we found that most of the critical points (normal-mild: 93%, mild- +moderate: 95%, moderate-advanced: 89%) were concentrated in the RNFL+PLT, sclera, and +LC. PointNet only focuses on the major structural changes of the optic nerve head, and since +we limited the number of critical points to 256, only the ONH landmarks with significant 3D +structural changes will be highlighted in the point cloud density maps. Therefore, the fact that +there are almost no critical points present in the GCC+IPL, ORL, RPE, and choroid does not +necessarily imply that these tissues do not exhibit any structural changes in glaucoma. +However, our findings suggest that any structural changes in these tissues are likely to be +smaller in magnitude compared to the structural changes observed in the RNFL, sclera, and +LC. + + +17 +In both approaches, we found that structural changes of neural tissues were more +prominent in the inferior and superior quadrants of the ONH over all stages of glaucoma. This +is in accordance with many previous studies (including our recent study in glaucoma diagnosis +[25]) that reported significant structural changes of glaucomatous ONHs in these quadrants +[30, 31]. In addition, Wang et al. reported a progressive nasalization of the central retinal +vessel trunk (CRVT) with glaucoma severity [32]. One might argue that the location of some +of the critical points from the RNFL coincides with the location of the CRVT and its branches +indicating changes in the CRVT location with disease progression. However, further research +is needed to confirm such speculations. +Furthermore, we found that the decline in MRW slowed, whereas RNFLT decreased +linearly as glaucoma severity increased. This suggests that neural tissue changes in the early +stage of glaucoma (normal to mild) are more pronounced around the optic disc (i.e. MRW), +in contrast to the later stages of glaucoma (mild to moderate and moderate to advanced), +where such changes move to the periphery of the ONH (i.e. RNFLT). Interestingly, we found a +similar trend in the distribution of critical points from the RNFL. In the early glaucoma group +(normal-mild), critical points were mostly located around the neuro-retinal rim. These critical +points (with their local tissue thickness) might act as a surrogate measurement for MRW. In +the more severe glaucoma groups (i.e. mild-moderate and moderate-advanced), critical +points from the RNFL moved to more peripheral regions of the ONH and thus closer to where +the RNFLT was measured. Up to date, there is no common consent on whether RNFLT or +MRW is better correlated with VF damage (i.e. glaucoma severity). Some studies favored +RNFLT [10, 33] whereas others reported better performance of MRW [30, 34]. In addition, +Gmeiner et al. reported that depending on the stage of glaucoma and the major site of +glaucomatous damage (peripheral or central), RNFLT might be superior to MRW and vice + + +18 +versa suggesting that morphological changes of the glaucomatous ONH are diverse and may +depend on various factors [33]. Therefore, when assessing ONH structural changes, it might +be important to analyze the entire region of the ONH (peripheral and central) with its complex +3D morphology as it was done with PointNet. +We found that a considerable number of critical points were extracted from the sclera +over all stages of glaucoma, suggesting significant and progressive structural changes of the +sclera with glaucoma severity. In addition, and in line with a previous study [17], we found +that the PPSA, representative for the bending of the sclera in the nasal-temporal plane, is +significantly larger in mild glaucoma compared to normal eyes, however, no significant +differences were found between the later stages of the disease. Considering the presence of +critical points from the sclera in all stages of glaucoma, one might speculate that a single +parameter like the PPSA is not enough to capture the complex 3D structural changes of the +sclera with glaucoma severity and further research is needed to quantify such changes. +Furthermore, we found that most of the LC critical points were located in the region of +the LC insertion zone over all stages of glaucoma. However, the major site of these critical +points changed from the superior quadrant (normal-mild) to the superior + inferior quadrant +(mild-moderate) to a more diffuse distribution over all quadrants (moderate-advanced). +Previous studies reported morphological changes of the LC with glaucoma severity reflected +by a change in LC depth [35], LC curvature [36], and LC-GSI [14]. In addition, local LC defects +or alterations like posterior movement of the LC insertion zones [37] and LC disinsertions [38] +were observed in glaucomatous eyes. However, none of the studies reported structural +changes of the LC insertion zone with glaucoma severity. Our findings suggest that assessing +morphological changes of the glaucomatous LC, especially in the region of the LC insertion +zone, could be useful in monitoring disease progression (in conjunction with other ONH + + +19 +parameters like the RNFLT). However, further longitudinal studies are necessary to unravel +the complex 3D structural changes of the LC with glaucoma severity. +In this study, several limitations warrant further discussion. First, although the overall +sample size was fairly large, however, subjects were unevenly distributed over the glaucoma +severity groups (normal: 213, mild: 204, moderate: 118, advanced: 118). In addition, the +Caucasian subgroup had no healthy controls that might introduce a bias in both, the +comparison of ONH parameters and the learning process of PointNet. Therefore, our findings +might not be easily transferable to other populations. In the future, we want to investigate +possible differences in structural changes of the ONH with glaucoma severity between +different ethnic groups. +Second, we used MD values of the 24-2 or 30-2 VF to determine glaucoma severity, +however, standard automated perimetry is subjective and sometimes underestimate disease +severity [22]. Recent studies suggest chromatic pupillometry [39] or electroretinogram [40] +as an objective way to assess functional loss in glaucomatous eyes. However, these devices +have their own limitations and a future study has to show whether our findings would change +when using a different staging system. +Third, the accuracy of the extracted ONH parameters and the extracted point clouds to +represent local structural features of the ONH depends on the performance of the +segmentation algorithm. Even though the segmentation software that we used in this study +(Reflectivity, Abyss Processing Pte Ltd, Singapore) was tested and validated on a large cohort +of glaucomatous and non-glaucomatous ONHs at different stages of glaucoma, one should +keep in mind that the choice of the segmentation algorithm might have an impact on the +results. + + +20 +Fourth, although we found that many ONH parameters showed significant differences +between glaucoma severity groups, the cross-sectional nature of our data limits causal +inferences. As a result, our findings might differ from longitudinal studies that follow +individual patients over a certain period of time. In the future, we aim to validate our findings +by applying our herein developed approaches to a longitudinal dataset. + Fifth, the differentiation of ONHs from the mild and moderate glaucoma severity group +was the most challenging task and resulted in a rather small AUC of 0.68  0.08 (PointNet). +The moderate performance of PointNet might be due to the plateau effect that we observed +after substantial structural changes in the early stage of glaucoma. In the future, we could +consider the MD value as a continuous variable and predict its “true” value, instead of a binary +classification, as this might give us a boost in performance. +In summary, we successfully described the 3D structural phenotype of glaucoma as a +function of glaucoma severity by: (1) a “traditional” approach based on extracted ONH +parameters and (2) a more recently introduced approach based on critical points extracted +by PointNet. We showed that ONH structural changes are not limited to neural tissues but +occurred in both, neural and connective tissues simultaneously. In addition, we identified a +major site of 3D morphological change of the ONH that might potentially be worth monitoring +in the future - the region around the LC insertion zone. With this study, we hope to provide +new insights into the complex pathophysiology of glaucoma that might help clinicians in their +daily clinical care. + +Acknowledgment + + +21 +We acknowledge funding from (1) the donors of the National Glaucoma Research, a +program of the BrightFocus Foundation, for support of this research (G2021010S [MJAG]); (2) +SingHealth Duke-NUS Academic Medicine Research Grant (SRDUKAMR21A6 [MJAG]); (3) the +“Retinal Analytics through Machine learning aiding Physics (RAMP)" project that is supported +by the National Research Foundation, Prime Minister’s Office, Singapore under its Intra- +Create Thematic Grant “Intersection Of Engineering And Health” - NRF2019-THE002-0006 +awarded to the Singapore MIT Alliance for Research and Technology (SMART) Centre +[MJAG/AT/GB]. (4) the “Tackling & Reducing Glaucoma Blindness with Emerging Technologies +(TARGET)” project that is supported by the National Medical Research Council (NMRC), +Singapore (MOH-OFLCG21jun-0003 [MJAG]). + + + + +22 +References + + +1. +Bussel, I.I., G. Wollstein, and J.S. Schuman, OCT for glaucoma diagnosis, screening +and detection of glaucoma progression. British Journal of Ophthalmology, 2014. +98(Suppl 2): p. ii15-ii19. +2. +Robin, T.A., et al., Performance of community-based glaucoma screening using +Frequency Doubling Technology and Heidelberg Retinal Tomography. Ophthalmic +Epidemiol, 2005. 12(3): p. 167-78. +3. +Lavinsky, F., et al., The Future of Imaging in Detecting Glaucoma Progression. +Ophthalmology, 2017. 124(12s): p. S76-s82. +4. +Gonzalez-Hernandez, M., et al., Structure–function relationship depends on +glaucoma severity. 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Biomed Opt +Express, 2020. 11(11): p. 6356-6378. +24. +Thakku, S.G., et al., A Global Shape Index to Characterize Anterior Lamina Cribrosa +Morphology and Its Determinants in Healthy Indian Eyes. Investigative +Ophthalmology & Visual Science, 2015. 56(6): p. 3604-3614. +25. +Braeu, F.A., et al. Geometric Deep Learning to Identify the Critical 3D Structural +Features of the Optic Nerve Head for Glaucoma Diagnosis. 2022. arXiv:2204.06931. +26. +Park, K.-H., et al., Bruch's membrane opening-minimum rim width and visual field +loss in glaucoma: a broken stick analysis. International journal of ophthalmology, +2018. 11(5): p. 828-834. +27. +Jonas, J.B. and A.E. Gründler, Correlation between mean visual field loss and +morphometric optic disk variables in the open-angle glaucomas. Am J Ophthalmol, +1997. 124(4): p. 488-97. +28. +Wollstein, G., et al., Retinal nerve fibre layer and visual function loss in glaucoma: the +tipping point. Br J Ophthalmol, 2012. 96(1): p. 47-52. +29. +Liebmann, K., C.G. De Moraes, and J.M. Liebmann, Measuring Rates of Visual Field +Progression in Linear Versus Nonlinear Scales: Implications for Understanding the +Relationship Between Baseline Damage and Target Rates of Glaucoma Progression. J +Glaucoma, 2017. 26(8): p. 721-725. +30. +Chauhan, B.C., et al., Enhanced Detection of Open-angle Glaucoma with an +Anatomically Accurate Optical Coherence Tomography–Derived Neuroretinal Rim +Parameter. Ophthalmology, 2013. 120(3): p. 535-543. +31. +Mwanza, J.-C., et al., Ability of Cirrus HD-OCT Optic Nerve Head Parameters to +Discriminate Normal from Glaucomatous Eyes. Ophthalmology, 2011. 118(2): p. 241- +248.e1. +32. +Wang, M., et al., Relationship Between Central Retinal Vessel Trunk Location and +Visual Field Loss in Glaucoma. American journal of ophthalmology, 2017. 176: p. 53- +60. +33. +Gmeiner, J.M.D., et al., Comparison of Bruch's Membrane Opening Minimum Rim +Width and Peripapillary Retinal Nerve Fiber Layer Thickness in Early Glaucoma +Assessment. Investigative Ophthalmology & Visual Science, 2016. 57(9): p. OCT575- +OCT584. + + +24 +34. +Muth, D.R. and C.W. Hirneiß, Structure–Function Relationship Between Bruch's +Membrane Opening–Based Optic Nerve Head Parameters and Visual Field Defects in +Glaucoma. Investigative Ophthalmology & Visual Science, 2015. 56(5): p. 3320-3328. +35. +Park, S.C., et al., Lamina cribrosa depth in different stages of glaucoma. Invest +Ophthalmol Vis Sci, 2015. 56(3): p. 2059-64. +36. +Lee, S.H., et al., Diagnostic Power of Lamina Cribrosa Depth and Curvature in +Glaucoma. Investigative Ophthalmology & Visual Science, 2017. 58(2): p. 755-762. +37. +Yang, H., et al., Posterior (outward) migration of the lamina cribrosa and early +cupping in monkey experimental glaucoma. Investigative ophthalmology & visual +science, 2011. 52(10): p. 7109-7121. +38. +Takayama, K., et al., Three-Dimensional Imaging of Lamina Cribrosa Defects in +Glaucoma Using Swept-Source Optical Coherence Tomography. Investigative +Ophthalmology & Visual Science, 2013. 54(7): p. 4798-4807. +39. +Najjar, R.P., et al., Handheld chromatic pupillometry can accurately and rapidly +reveal functional loss in glaucoma. British Journal of Ophthalmology, 2021: p. +bjophthalmol-2021-319938. +40. +Sarossy, M., et al., Prediction of glaucoma severity using parameters from the +electroretinogram. Scientific Reports, 2021. 11(1): p. 23886. + + + + + +25 +Figures + + + +Figure 1. Overview of two approaches to describe the 3D structural phenotype of glaucoma +as a function of severity. Approach 1 was based on the comparison of well-established ONH +parameters between different glaucoma severity groups (a-c). Approach 2 leverages on + + +26 +geometric deep learning to identify important 3D landmarks of the ONH to differentiate ONHs +at different stages of glaucoma. By looking at the changes of these critical 3D structural +features with glaucoma severity, we were able to draw conclusions about the complex 3D +structural changes of the ONH taking place at different stages of glaucoma (a, b, d, and e). + + + + +27 + + + +28 +Figure 2. Summary of statistical analysis of automatically extracted ONH parameters. RNFLT, +MRW, GCCT, and ChT are shown as sector plots (T: temporal, ST: superior-temporal, S: +superior, SN: superior-nasal, N: nasal, NI: nasal-inferior, and I: inferior sector) with values for +each group given as average  standard deviation. Non-sectorial parameters are presented +as boxplots. A significant difference between two groups (p<0.05) was indicated with a * +(determined by post-hoc Tukey HSD tests). + + + + + +Figure 3. Critical points resulting from the three classification tasks: normal-mild, mild- +moderate, and moderate advanced. From left to right column: 3D, en face (top), and sagittal +(side) view. Surfaces represent the average anterior tissue boundaries for each respective +dataset: RNFL+PLT (red), GCL+IPL (green), ORL (blue), RPE (yellow), choroid (purple), sclera +(cyan), and LC (orange). Red colored critical points correspond to ONH regions with high +importance for the differentiation of the respective glaucoma severity groups. + + + +29 + +1 + +2 +Figure 4. En face (top) view layer by layer comparison (columns) of critical points at different stages of glaucoma severity (rows). Critical points +3 +are presented as point cloud density maps with colours indicating the number of neighbouring points within a sphere with a radius of 75 µm. + +4 + + +30 +Tables +5 + +6 +Table 1. Summary of glaucoma severity groups. +7 + +8 + +NORMAL +(N=213) +MILD +(N=204) +MODERATE +(N=118) +ADVANCED +(N=118) +P* +AGE, YEARS +63.36 (6.99) +66.9 (6.42) +68.05 (7.11) +68.52 (7.69) +<0.001 +SEX, FEMALE +126 (59.15) +91 (44.61) +49 (41.52) +43 (36.44) +<0.001 +RACE + + + + + + CHINESE +213 +178 +97 +53 +<0.001 + CAUCASIAN +0 +26 +21 +65 +MD, DB +-1.41 (2.11) +-3.35 (1.95) +-8.16 (2.35) +-18.64 (5.31) +<0.001 +Data are in mean (standard deviation) or n (%) as appropriate. +9 +MD = mean deviation of the 24-2 or 30-2 visual field test. +10 +*Comparison between the four groups using Fisher’s exact test (for sex and race) and ANOVA +11 +(for age and MD). +12 + diff --git a/3NE1T4oBgHgl3EQfAQKK/content/tmp_files/load_file.txt b/3NE1T4oBgHgl3EQfAQKK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3735d603a480ee9319d39c1d74e92bd7424cdb78 --- /dev/null +++ b/3NE1T4oBgHgl3EQfAQKK/content/tmp_files/load_file.txt @@ -0,0 +1,679 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf,len=678 +page_content='1 The 3D Structural Phenotype of the Glaucomatous Optic Nerve Head and its Relationship with The Severity of Visual Field Damage Fabian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Braeu1,2,3, Thanadet Chuangsuwanich1,3, Tin A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Tun4,5, Shamira A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Perera4,5, Rahat Husain4, Aiste Kadziauskiene6,7, Leopold Schmetterer4,5,8-12, Alexandre H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Thiéry13, George Barbastathis2,14, Tin Aung3,4,5, and Michaël J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Girard1,5,12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Ophthalmic Engineering & Innovation Laboratory, Singapore Eye Research Institute, Singapore National Eye Centre, Singapore 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Singapore-MIT Alliance for Research and Technology, Singapore 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Yong Loo Lin School of Medicine, National University of Singapore, Singapore 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Singapore Eye Research Institute, Singapore National Eye Centre, Singapore 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Duke-NUS Graduate Medical School, Singapore 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Clinic of Ears, Nose, Throat and Eye Diseases, Institute of Clinical Medicine, Faculty of Medicine, Vilnius University, Vilnius, Lithuania 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Center of Eye Diseases, Vilnius University Hospital Santaros Klinikos, Vilnius, Lithuania 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' SERI-NTU Advanced Ocular Engineering (STANCE), Singapore, Singapore 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' School of Chemistry, Chemical Engineering and Biotechnology, Nanyang Technological University Singapore 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Department of Clinical Pharmacology, Medical University of Vienna, Austria 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Center for Medical Physics and Biomedical Engineering, Medical University of Vienna, Austria 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Institute of Molecular and Clinical Ophthalmology, Basel, Switzerland 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Department of Statistics and Applied Probability, National University of Singapore, Singapore 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Keywords: Geometric deep learning, glaucoma, artificial intelligence, optic nerve head, PointNet Word count: 4,971 (Manuscript Text) 339 (Abstract) Tables: 1 Figures: 4 Conflict of Interest: MJAG and AHT are the co-founders of the AI start-up company Abyss Processing Pte Ltd Corresponding Author: Michaël J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Girard Ophthalmic Engineering & Innovation Laboratory (OEIL) Singapore Eye Research Institute (SERI) The Academia, 20 College Road Discovery Tower Level 6, Singapore 169856 mgirard@ophthalmic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='engineering https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='ophthalmic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='engineering 2 Abstract Purpose: To describe the 3D structural changes in both connective and neural tissues of the optic nerve head (ONH) that occur concurrently at different stages of glaucoma using traditional and AI-driven approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Design: Retrospective cross-sectional study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Methods: We included 213 normal, 204 mild glaucoma (mean deviation [MD] ≥ -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='00 dB), 118 moderate glaucoma (MD of -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='01 to -12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='00 dB), and 118 advanced glaucoma patients (MD < -12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='00 dB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' All subjects had their ONHs imaged in 3D with Spectralis optical coherence tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' To describe the 3D structural phenotype of glaucoma as a function of severity, we used two different approaches: (1) We extracted ‘human-defined’ 3D structural parameters of the ONH (total of 10) including retinal nerve fiber layer (RNFL) thickness, minimum rim width, lamina cribrosa (LC) shape and depth at different stages of glaucoma;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' (2) we also employed a geometric deep learning method (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' PointNet) to identify the most important 3D structural features that differentiate ONHs from different glaucoma severity groups without any human input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Results: We observed that the majority of ONH structural changes occurred in the early glaucoma stage, followed by a plateau effect in the later stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Using PointNet, we also found that 3D ONH structural changes were present in both neural and connective tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Specifically, 57% (normal to mild glaucoma), 39% (mild to moderate glaucoma), and 53% (moderate to advanced glaucoma) of ONH landmarks that showed major structural changes were located in neural tissues with the remaining located in connective tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In both approaches, we observed that structural changes were more prominent in the superior and inferior quadrant of the ONH, particularly in the RNFL, the prelamina, and the LC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' As the 3 severity of glaucoma increased, these changes became more diffuse (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' widespread), particularly in the LC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Conclusions: In this study, we were able to uncover complex 3D structural changes of the ONH in both neural and connective tissues as a function of glaucoma severity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' We hope to provide new insights into the complex pathophysiology of glaucoma that might help clinicians in their daily clinical care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' 4 Introduction Evaluation of structural changes of the optic nerve head (ONH) – the main site of damage in glaucoma – is a crucial step in diagnosing and monitoring glaucoma [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' The complex three-dimensional (3D) morphological changes occurring in glaucomatous ONHs can be captured and quantified by optical coherence tomography (OCT) – a fast, high-resolution, quantitative, and non-invasive 3D imaging modality [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In current medical practice, several investigations are conducted to assess neural tissue health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' These tests involve both a functional (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=', visual field testing) and a structural assessment of glaucomatous damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' The latter is typically achieved by measuring the thickness of the retinal nerve fiber layer (RNFL) via OCT [4-6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Researchers have further investigated the association between other neural structural parameters with glaucomatous visual field damage, such as the thickness of the ganglion cell complex (GCC) [7, 8] and Bruch’s membrane opening - minimum rim width (BMO-MRW) [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' However, recent research has indicated that the pathophysiology of glaucoma is multifaceted and cannot purely be characterized as damage to retinal ganglion cells: (1) Brooks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' reported that the characteristic “glaucomatous cupping” of the ONH cannot solely be explained by neural tissue loss [11];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' (2) Quigley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' found that glaucomatous changes of the lamina cribrosa (LC) precede visual field damage [12];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' and (3) Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' suggested that ONH connective tissue deformations are the primary cause of retinal ganglion cell axonal injury [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' These studies indicate that the pathophysiology of glaucoma should consider the involvement of the biomechanics, mechanobiology, remodeling, and potential mechanical breakdown of ONH connective tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Given this new understanding, researchers have begun to investigate the association between ONH connective tissue changes and 5 glaucoma severity through connective tissue parameters extracted from OCT images of the ONH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Examples of such parameters include the LC depth (LCD) and the LC global shape index (LC-GSI) [14], the thickness of the peripapillary choroid [15], the scleral canal opening [16], and the peripapillary scleral angle representing the amount of bowing of the ONH [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' However, no study has yet provided a comprehensive analysis of 3D structural changes of both the connective and neural tissues of the ONH that occur concurrently at different stages of glaucoma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Therefore, the aim of this study was to describe the 3D structural phenotype of glaucoma as a function of severity by: (1) Extracting neural and connective tissue ONH parameters from segmented 3D OCT scans and investigating their differences between glaucoma severity groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' (2) Using 3D point clouds representing the complex structure of the ONH as an input for a geometric deep learning technique (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' PointNet [18]) that allows us to identify the major 3D structural changes of the ONH with glaucoma severity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Overall, we hope that our work leads to a better understanding of the pathophysiology of glaucoma that might improve the diagnosis and prognosis of glaucoma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Methods Patient Recruitment This retrospective study involved a total of 414 subjects with glaucoma and 213 controls without glaucoma from two different cohorts: (1) 541 subjects of Chinese ethnicity were recruited at the Singapore National Eye Centre (SNEC) as part of their standard clinical care and (2) 112 subjects of European descent were recruited at the Vilnius University Hospital Santaros Klinikos as part of a prospective observational study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' All subjects gave 6 written informed consent and the study adhered to the tenets of the Declaration of Helsinki and was approved by the institutional review board of the respective institutions (SingHealth Centralized Institutional review board, Singapore and Vilnius Regional Biomedical Research Ethics Committee, Lithuania).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Standard Automated Perimetry All subjects had their visual field (VF) assessed by standard automated perimetry (SAP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Swedish Interactive Threshold Algorithm standard 24-2 or 30-2 program;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Humphrey Field Analyzer II-750i, Carl Zeiss Meditec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Subjects with a non-reliable VF examination that was defined using the criteria of a false-positive error rate greater than 15% [19] and a fixation loss greater than 33% [19, 20] were excluded from the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Definition of Glaucoma and Glaucoma Severity Groups Glaucomatous eyes were defined as those with vertical cup-disc ratio (VCDR) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='7 and/or neuroretinal rim narrowing with repeatable glaucomatous VF defects and non- occludable angles on gonioscopy whereas non-glaucomatous (normal) eyes were those with an IOP < 21 mmHg and normal VF examinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Subjects with corneal abnormalities that potentially can reduce the quality of the OCT scans and with ONH disorders other than glaucoma were excluded from the studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Based upon the mean deviation (MD) of the 24-2 or 30-2 VF, all glaucoma subjects were further split into three glaucoma severity groups [21]: (1) mild glaucoma (MD ≥ -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='00 dB);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' (2) moderate glaucoma (MD of -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='01 to -12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='00 dB);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' and (3) advanced glaucoma (MD < - 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='00 dB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Even though this classification has its limitations [22], it remains a standard and can be used as a good first indicator for staging functional damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' More information on the 7 demographics of the four groups (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' normal, mild, moderate, and advanced) can be found in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Optical Coherence Tomography Imaging Each patient from both cohorts had their ONH imaged with the same spectral domain OCT device (Spectralis, Heidelberg Engineering, Germany).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' All OCT scans (horizontal raster scans) covered an area of 15\uf0b0x10\uf0b0 centered on the ONH and the number of B-scans varied between 49 and 97 B-scans (distance between B-scans varied from approximately 35 to 70 µm) with 384 A-scans per B-scan (approximately 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='5 µm between A-scans) and 496 pixels per A-scan (axial resolution of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='87 µm/pixel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Images were acquired using signal averaging, eye tracking, and the enhanced depth imaging modality of the Spectralis OCT device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Describing the Structural Phenotype of Glaucoma as a Function of Glaucoma Severity In the following sections, we introduce two different approaches to study the complex structural changes of the ONH as a function of glaucoma severity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In the first, we performed a comprehensive 3D structural analysis of the ONH using ‘human-defined’ 3D structural parameters of the ONH (total of 10 parameters) describing the morphologies of both neural and connective tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In the second, we used a relatively recent geometric deep learning method (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' PointNet) to discover important 3D structural features differentiating ONHs from different glaucoma severity groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' An overview of both approaches is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Approach 1 for Describing the Structural Phenotype of Glaucoma – ONH Parameters For this approach, all ONH tissues were segmented in 3D (from the OCT scans), from which all ONH structural parameters were automatically extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' 8 AI-based Segmentation of ONH Tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' We automatically segmented all raw OCT volume scans of the ONH using REFLECTIVITY (Reflectivity, Abyss Processing Pte Ltd, Singapore) – a software that was developed from advances in AI-based ONH segmentation [23] (see Figure 1a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' More specifically, we automatically labelled the following ONH tissue groups: (1) the retinal nerve fiber layer (RNFL) and the prelamina tissue (PLT);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' (2) the ganglion cell inner plexiform layer (GCL+IPL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' (3) all other retinal layers (ORL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' (4) the retinal pigment epithelium (RPE) with Bruch’s membrane (BM) and the BM opening (BMO) points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' (5) the choroid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' (6) the OCT-visible part of the peripapillary sclera including the scleral flange;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' and (7) the OCT-visible part of the LC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In almost all OCT volume scans, the posterior boundaries of the sclera and LC were not visible and could therefore not be segmented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Automated extraction of ONH parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Using the software REFLECTIVITY, we extracted the following parameters: (1) the average RNFL thickness (RNFLT) in each octant (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' temporal [T], superior-temporal [ST], superior [S], superior-nasal [SN], nasal [N], inferior- nasal [IN], inferior [I], and inferior-temporal [IT]) calculated at a distance of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='5 times the BMO radius (BMOR) from the centre of BMO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' (2) the average minimum rim width (MRW) in each octant defined as the minimum distance from a BMO point to a point on the inner limiting membrane (ILM);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' (3) the average ganglion cell complex thickness (GCCT) in each octant evaluated at the same location than the RNFLT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' (4) the average choroidal thickness (ChT) in each octant at the same distance as that used for the RNFLT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' (5) the prelamina depth (PLD) defined as the distance from the BMO center to a point on the ILM (perpendicular to the BMO plane);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' (6) the minimum prelamina thickness (MPT);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' (7) the LC depth (LCD) defined as the distance from the BMO centre to a point on the anterior LC boundary (perpendicular to the BMO plane);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' (8) the LC global shape index (LC-GSI) that summarizes the shape of the anterior LC boundary into a single number [24];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' (9) the peripapillary scleral angle (PPSA) representing 9 the amount of scleral bowing and defined as the angle between two parallel lines to the anterior scleral boundary in the nasal-temporal plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' and (10) the BMO area defined as the area of the best-fit ellipse to the BMO points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' A visualization of the extracted ONH parameters is shown in Figure 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Statistical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' All parameters were compared across all 4 groups (normal, mild, moderate, and advanced).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' All statistical analyses were performed using R (version 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='1) and RStudio (version 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='1 for macOS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' ONH parameters that were extracted in each octant were reported as mean \uf0b1 standard deviation and single valued ONH parameters were presented as box plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' One-way ANOVA with post-hoc Tukey HSD test was used for the comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' P value for significance was set at <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Approach 2 for Describing the Structural Phenotype of Glaucoma – PointNet PointNet, a deep neural network from the group of geometric deep learning algorithms, can learn from complex 3D shapes, such as that of the ONH, if they are represented as 3D point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In contrast to our first approach, which relied on ‘human- defined’ ONH parameters, PointNet allows us to identify important structural landmarks that can differentiate ONHs from the four different glaucoma severity groups, without previous inputs or guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Representation of the ONH structure as 3D point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' We described the structure of a given ONH as a 3D point cloud which then was used as input to PointNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' To do so, we first identified the anterior boundaries of all tissue layers in the segmented OCT scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Each anterior boundary voxel was then represented as a 3D point (see Figure 1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' The final point cloud consisted of about 20,000 points for each ONH (see Figure 1e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Additionally, for each point, we extracted the local tissue thickness (minimum distance between anterior and posterior boundary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In summary, we assigned four values to every point: its position in the 10 3D space ([x, y, z]-coordinate) and its local tissue thickness (not applicable for the sclera and LC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' To homogenize the data across all ONHs, the centre of BMO was set as origin of the coordinate system [x=0, y=0, z=0] and the normal of BMO plane (best-fit plane to the BMO points) was aligned with the axial direction of the scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' The more interested reader is referred to our previous publication on geometric deep learning for glaucoma diagnosis [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Glaucoma severity classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' PointNet was specifically designed to process and learn from 3D point clouds such as the one shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' We used the same architecture as in the original publication [18], except that we implemented a max pooling layer of dimension 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' To identify important 3D structural features of the ONH at different stages of glaucoma, we trained three PointNet classification networks to differentiate between: (1) normal and mild glaucoma subjects (normal-mild);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' (2) mild and moderate glaucoma subjects (mild-moderate);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' and (3) moderate and advanced glaucoma subjects (moderate-advanced).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' To assess the performance of the three binary classification networks, we split each respective dataset (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' normal-mild, mild-moderate, and moderate-advanced) in training (70%), validation (15%), and test (15%) sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' To improve performance and reduce overfitting, we used data augmentation techniques such as random cropping, random rotations, random rigid translations, random sampling (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' randomly picking a subset of points from the input point cloud), oversampling to reduce data imbalance, and additive Gaussian noise where applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' A five-fold cross validation study was performed (using the train and validation set) to tune hyperparameters and we reported the area under the receiver operating characteristic curves (AUCs) of the model with the best performing hyperparameters as mean ± standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' All models were trained on a Nvidia RTX A5000 GPU card until optimum performance was reached in the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' 11 Identification of important 3D structural features of the ONH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' The specific architecture of PointNet inherently allowed us to identify regions of the ONH important for the differentiation of different glaucoma severity groups by extracting all points that contributed to the final classification score – the so-called critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' For each classification group (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' normal-mild, mild-moderate, and moderate-advanced), we extracted critical points from all ONHs of the respective test set (networks trained on respective training set using tuned hyperparameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Comparing the locations of these points between the three groups allowed us to draw conclusion on the characteristic 3D structural changes of the ONH at different stages of glaucoma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Visualization of critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' To better visualize the location of the resulting critical points, we first constructed an average ONH geometry (represented by the average anterior boundaries of each segmented tissue) for each of the three classification groups, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' normal- mild, mild-moderate, and moderate-advanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' For each group, we then projected the critical points (closest point projection) onto their corresponding anterior tissue boundary of the respective average ONH geometry and visualized them as 3D point cloud density maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' A density measure for each point was obtained by counting the neighbouring points within a 75 μm radius sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Since all critical points were projected on an average ONH geometry, such a density map should highlight landmarks of the ONH that exhibit distinct 3D structural changes between the different stages of glaucoma (represented as a cluster of red points in the point cloud density maps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' 12 Results Approach 1 – Statistical Analysis of ONH Parameters We observed that the majority of ONH structural changes occurred in the early glaucoma stage (normal to mild).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' These changes were also the most substantial in terms of their size or magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Specifically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' we noted a decrease in average RNFLT (average over all sectors) from 112 \uf0b1 26 µm to 83 \uf0b1 29 µm (Figure 2a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' a decrease in average MRW from 256 \uf0b1 60 µm to 169 \uf0b1 55 µm (Figure 2b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' a decrease in average GCCT from 154 \uf0b1 26 µm to 124 \uf0b1 30 µm (Figure 2c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' no change in average ChT (Figure 2d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' an increase in PLD from 136 \uf0b1 195 µm to 288 \uf0b1 199 µm (Figure 2e),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' a decrease in MPT from 146 \uf0b1 116 µm to 63 \uf0b1 70 µm (Figure 2f),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' an increase in LCD from 410 \uf0b1 109 µm to 468 \uf0b1 132 µm (Figure 2g),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' a decrease in LC-GSI from -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='37 \uf0b1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='42 to -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='61 \uf0b1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='33 (Figure 2h), an increase in PPSA from 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='4 \uf0b1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='6 degree to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='5 \uf0b1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='2 degree (Figure 2i), and an increase in BMOA from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='15 \uf0b1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='5 mm2 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='28 \uf0b1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='5 mm2 (Figure 2j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Following substantial structural changes of the ONH in the early stage of glaucoma, most ONH parameters showed a plateau effect, with little change from mild to moderate glaucoma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Only RNFLT (average), GCCT (average), and MRW (average) showed a significant decrease from 83 \uf0b1 29 to 71 \uf0b1 30 µm, 124 \uf0b1 30 to 111 \uf0b1 32 µm, and 169 \uf0b1 55 to 159 \uf0b1 56 µm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In the later stages of glaucoma (moderate to advanced), we observed significant structural changes of the ONH, but they were much less pronounced in terms of their magnitude compared to those seen in the early stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In detail, the average RNFLT decreased from 71 \uf0b1 30 µm to 50 \uf0b1 25 µm (Figure 2a), the average MRW decreased from 159 \uf0b1 56 µm to 126 \uf0b1 46 µm (Figure 2b), the average GCCT decreased from 111 \uf0b1 32 µm to 88 \uf0b1 27 µm 13 (Figure 2c), the LCD increased from 459 \uf0b1 121 to 502 \uf0b1 147 µm (Figure 2g), and the BMOA decreased from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='30 \uf0b1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='58 mm2 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='12 \uf0b1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='42 mm2 (Figure 2j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' The ChT (Figure 2d), the PLD (Figure 2e), the MPT (Figure 2f), the LC-GSI (Figure 2h), and the PPSA (Figure 2i) showed no significant change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' If we were to examine regional variations, we noted that structural changes of the RNFLT, MRW, and GCCT were more pronounced (higher in magnitude) in both the superior and inferior octants of the ONH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' This was true throughout all stages of glaucoma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In these sectors, we observed that the decrease in MRW slowed as glaucoma severity increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Specifically, in the early stage of glaucoma (normal to mild), MRW decreased in the superior octant from 295 \uf0b1 64 µm to 192 \uf0b1 58 µm while in the later stage (moderate to advanced), the decrease was smaller from 179 \uf0b1 58 µm to 133 \uf0b1 49 µm (Figure 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In contrast, RNFLT and GCCT decreased linearly as glaucoma severity increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In the early stage of glaucoma (normal to mild), RNFLT and GCCT in the superior octant decreased from 163 \uf0b1 31 to 122 \uf0b1 34 µm and 200 \uf0b1 31 to 160 \uf0b1 34 µm, respectively, while in the later stage (moderate to advanced), the decrease was from 102 \uf0b1 35 to 61 \uf0b1 31 µm and 141 \uf0b1 35 to 99 \uf0b1 32 µm (Figure 2a, 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' With the exception of the inferior octant of the ONH, we did not observe any significant changes in the ChT with glaucoma severity (Figure 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Approach 2 – Performance Assessment Using PointNet, we were able to differentiate ONHs from different glaucoma severity groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' The normal-mild glaucoma classification showed the best performance (AUC: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='94 \uf0b1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='02), followed by the moderate-advanced (AUC: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='80 \uf0b1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='04) and mild-moderate glaucoma classification (AUC: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='68 \uf0b1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='08).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' 14 Approach 2 – Changes of Important 3D Structural Features of the ONH with Glaucoma Severity For each classification task (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' normal-mild, mild-moderate, and moderate- advanced), we pooled all critical points from all ONHs (test set), mapped them onto the corresponding average ONH geometry, and displayed them as a 3D point cloud density map for all ONH tissues (Figure 3), or separately for each ONH tissue (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In general, we observed that critical points were present in both, neural (normal-mild: 57%, mild-moderate: 39%, moderate-advanced: 53%) and connective tissues (normal-mild: 43%, mild-moderate: 61%, moderate-advanced: 47%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' More specifically, most of the critical points were located in the RNFL+PLT (normal-mild: 53%, mild-moderate: 37%, moderate- advanced: 47%), the sclera (normal-mild: 17%, mild-moderate: 15%, moderate-advanced: 11%), and the LC (normal-mild: 23%, mild-moderate: 43%, moderate-advanced: 31%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In contrast, we observed almost no critical points in the other tissue layers, such as the GCC+IPL, ORL, RPE, Choroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' On a tissue level, we found that the critical points from the RNFL of all three classification tasks formed an hourglass pattern with points mainly located in the superior and inferior quadrant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In addition, in the normal-mild glaucoma classification, critical points from the RNFL were mostly located around the neuro-retinal rim whereas in the moderate- advanced glaucoma classification, these points moved more outwards to the peripheral region of the ONH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Interestingly, we also found that in the normal-mild and mild-moderate classification most of the critical points from the LC were located near the LC insertion zone in the superior (normal-mild) and superior + inferior quadrant (mild-moderate) whereas in 15 the moderate-advanced classification, critical points were more spread out over the entire LC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Discussion In this study, we were able to describe the 3D structural phenotype of glaucoma as a function of severity using two separate approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In the first, we extracted ‘human-defined’ 3D structural parameters of the ONH and compared them across four different groups: normal, mild, moderate, and advanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In the second, we represented the complex structure of the ONH as a 3D point cloud and used PointNet to uncover the structural landmarks that were the most affected by glaucoma severity without any human input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Overall, we found that the structural features of both neural and connective tissues contributed to the structural phenotype of glaucoma, and that each of our proposed method could provide its own unique knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In this study, we found that after substantial structural changes of the ONH in the early stage of glaucoma (normal to mild), almost all ONH parameters reached a plateau, with less change in the later stages (mild to moderate and moderate to advanced).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' This is in good agreement with previous studies that investigated the structure-function relationship and reported a considerable structural loss before any functional VF defects were detectable [26- 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Some of these studies suggested a “tipping point” in the early stage of glaucoma (at about – 3 dB MD) from which onwards even small structural changes were associated with a relatively large decrease in MD value [26, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' One should also keep in mind that MD values are usually reported on a logarithmic scale (non-linear scale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' For instance, a shift in MD value from 0 to -6 dB would imply a much larger loss in visual sensitivity compared to a shift from - 16 6 to -12 dB on a linear scale [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Therefore, the observed plateau effect might be a result of reporting MD values on a logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' However, further research is needed to verify such a hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Furthermore, we found that critical points were present in both neural (normal-mild: 57%, mild-moderate: 39%, moderate-advanced: 53%) and connective tissues (normal-mild: 43%, mild-moderate: 61%, moderate-advanced: 47%) at all stages of glaucoma indicating that the structural changes caused by glaucoma affected both types of tissue in the ONH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Our findings are in line with previous research that suggested that the pathophysiology of glaucoma is complex and cannot purely be characterized as a damage to the neural tissue in the ONH (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' retinal ganglion cells) [11-13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Despite these recent findings, current glaucoma tests focus on assessing neural tissue health, ignoring any glaucomatous structural changes of connective tissue in the ONH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In the future, the development of more comprehensive tests that consider structural changes in both, neural and connective tissues, could potentially improve the diagnosis and prognosis of glaucoma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Additionally, we found that most of the critical points (normal-mild: 93%, mild- moderate: 95%, moderate-advanced: 89%) were concentrated in the RNFL+PLT, sclera, and LC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' PointNet only focuses on the major structural changes of the optic nerve head, and since we limited the number of critical points to 256, only the ONH landmarks with significant 3D structural changes will be highlighted in the point cloud density maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Therefore, the fact that there are almost no critical points present in the GCC+IPL, ORL, RPE, and choroid does not necessarily imply that these tissues do not exhibit any structural changes in glaucoma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' However, our findings suggest that any structural changes in these tissues are likely to be smaller in magnitude compared to the structural changes observed in the RNFL, sclera, and LC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' 17 In both approaches, we found that structural changes of neural tissues were more prominent in the inferior and superior quadrants of the ONH over all stages of glaucoma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' This is in accordance with many previous studies (including our recent study in glaucoma diagnosis [25]) that reported significant structural changes of glaucomatous ONHs in these quadrants [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In addition, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' reported a progressive nasalization of the central retinal vessel trunk (CRVT) with glaucoma severity [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' One might argue that the location of some of the critical points from the RNFL coincides with the location of the CRVT and its branches indicating changes in the CRVT location with disease progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' However, further research is needed to confirm such speculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Furthermore, we found that the decline in MRW slowed, whereas RNFLT decreased linearly as glaucoma severity increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' This suggests that neural tissue changes in the early stage of glaucoma (normal to mild) are more pronounced around the optic disc (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' MRW), in contrast to the later stages of glaucoma (mild to moderate and moderate to advanced), where such changes move to the periphery of the ONH (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' RNFLT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Interestingly, we found a similar trend in the distribution of critical points from the RNFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In the early glaucoma group (normal-mild), critical points were mostly located around the neuro-retinal rim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' These critical points (with their local tissue thickness) might act as a surrogate measurement for MRW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In the more severe glaucoma groups (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' mild-moderate and moderate-advanced), critical points from the RNFL moved to more peripheral regions of the ONH and thus closer to where the RNFLT was measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Up to date, there is no common consent on whether RNFLT or MRW is better correlated with VF damage (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' glaucoma severity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Some studies favored RNFLT [10, 33] whereas others reported better performance of MRW [30, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In addition, Gmeiner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' reported that depending on the stage of glaucoma and the major site of glaucomatous damage (peripheral or central), RNFLT might be superior to MRW and vice 18 versa suggesting that morphological changes of the glaucomatous ONH are diverse and may depend on various factors [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Therefore, when assessing ONH structural changes, it might be important to analyze the entire region of the ONH (peripheral and central) with its complex 3D morphology as it was done with PointNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' We found that a considerable number of critical points were extracted from the sclera over all stages of glaucoma, suggesting significant and progressive structural changes of the sclera with glaucoma severity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In addition, and in line with a previous study [17], we found that the PPSA, representative for the bending of the sclera in the nasal-temporal plane, is significantly larger in mild glaucoma compared to normal eyes, however, no significant differences were found between the later stages of the disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Considering the presence of critical points from the sclera in all stages of glaucoma, one might speculate that a single parameter like the PPSA is not enough to capture the complex 3D structural changes of the sclera with glaucoma severity and further research is needed to quantify such changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Furthermore, we found that most of the LC critical points were located in the region of the LC insertion zone over all stages of glaucoma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' However, the major site of these critical points changed from the superior quadrant (normal-mild) to the superior + inferior quadrant (mild-moderate) to a more diffuse distribution over all quadrants (moderate-advanced).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Previous studies reported morphological changes of the LC with glaucoma severity reflected by a change in LC depth [35], LC curvature [36], and LC-GSI [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In addition, local LC defects or alterations like posterior movement of the LC insertion zones [37] and LC disinsertions [38] were observed in glaucomatous eyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' However, none of the studies reported structural changes of the LC insertion zone with glaucoma severity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Our findings suggest that assessing morphological changes of the glaucomatous LC, especially in the region of the LC insertion zone, could be useful in monitoring disease progression (in conjunction with other ONH 19 parameters like the RNFLT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' However, further longitudinal studies are necessary to unravel the complex 3D structural changes of the LC with glaucoma severity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In this study, several limitations warrant further discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' First, although the overall sample size was fairly large, however, subjects were unevenly distributed over the glaucoma severity groups (normal: 213, mild: 204, moderate: 118, advanced: 118).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In addition, the Caucasian subgroup had no healthy controls that might introduce a bias in both, the comparison of ONH parameters and the learning process of PointNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Therefore, our findings might not be easily transferable to other populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In the future, we want to investigate possible differences in structural changes of the ONH with glaucoma severity between different ethnic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Second, we used MD values of the 24-2 or 30-2 VF to determine glaucoma severity, however, standard automated perimetry is subjective and sometimes underestimate disease severity [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Recent studies suggest chromatic pupillometry [39] or electroretinogram [40] as an objective way to assess functional loss in glaucomatous eyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' However, these devices have their own limitations and a future study has to show whether our findings would change when using a different staging system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Third, the accuracy of the extracted ONH parameters and the extracted point clouds to represent local structural features of the ONH depends on the performance of the segmentation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Even though the segmentation software that we used in this study (Reflectivity, Abyss Processing Pte Ltd, Singapore) was tested and validated on a large cohort of glaucomatous and non-glaucomatous ONHs at different stages of glaucoma, one should keep in mind that the choice of the segmentation algorithm might have an impact on the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' 20 Fourth, although we found that many ONH parameters showed significant differences between glaucoma severity groups, the cross-sectional nature of our data limits causal inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' As a result, our findings might differ from longitudinal studies that follow individual patients over a certain period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In the future, we aim to validate our findings by applying our herein developed approaches to a longitudinal dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Fifth, the differentiation of ONHs from the mild and moderate glaucoma severity group was the most challenging task and resulted in a rather small AUC of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='68 \uf0b1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='08 (PointNet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' The moderate performance of PointNet might be due to the plateau effect that we observed after substantial structural changes in the early stage of glaucoma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In the future, we could consider the MD value as a continuous variable and predict its “true” value, instead of a binary classification, as this might give us a boost in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In summary, we successfully described the 3D structural phenotype of glaucoma as a function of glaucoma severity by: (1) a “traditional” approach based on extracted ONH parameters and (2) a more recently introduced approach based on critical points extracted by PointNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' We showed that ONH structural changes are not limited to neural tissues but occurred in both, neural and connective tissues simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' In addition, we identified a major site of 3D morphological change of the ONH that might potentially be worth monitoring in the future - the region around the LC insertion zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' With this study, we hope to provide new insights into the complex pathophysiology of glaucoma that might help clinicians in their daily clinical care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Acknowledgment 21 We acknowledge funding from (1) the donors of the National Glaucoma Research, a program of the BrightFocus Foundation, for support of this research (G2021010S [MJAG]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' (2) SingHealth Duke-NUS Academic Medicine Research Grant (SRDUKAMR21A6 [MJAG]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' (3) the “Retinal Analytics through Machine learning aiding Physics (RAMP)" project that is supported by the National Research Foundation, Prime Minister’s Office, Singapore under its Intra- Create Thematic Grant “Intersection Of Engineering And Health” - NRF2019-THE002-0006 awarded to the Singapore MIT Alliance for Research and Technology (SMART) Centre [MJAG/AT/GB].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' (4) the “Tackling & Reducing Glaucoma Blindness with Emerging Technologies (TARGET)” project that is supported by the National Medical Research Council (NMRC), Singapore (MOH-OFLCG21jun-0003 [MJAG]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' 22 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Bussel, I.' metadata={'source': 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and Curvature in Glaucoma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Investigative Ophthalmology & Visual Science, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' 58(2): p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' 755-762.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Yang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=', et al.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Najjar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=', Handheld chromatic pupillometry can accurately and rapidly reveal functional loss in glaucoma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' British Journal of Ophthalmology, 2021: p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' bjophthalmol-2021-319938.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Sarossy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=', Prediction of glaucoma severity using parameters from the electroretinogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Scientific Reports, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' 11(1): p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' 23886.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' 25 Figures Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Overview of two approaches to describe the 3D structural phenotype of glaucoma as a function of severity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Approach 1 was based on the comparison of well-established ONH parameters between different glaucoma severity groups (a-c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Approach 2 leverages on 26 geometric deep learning to identify important 3D landmarks of the ONH to differentiate ONHs at different stages of glaucoma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' By looking at the changes of these critical 3D structural features with glaucoma severity, we were able to draw conclusions about the complex 3D structural changes of the ONH taking place at different stages of glaucoma (a, b, d, and e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' 27 28 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Summary of statistical analysis of automatically extracted ONH parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' RNFLT, MRW, GCCT, and ChT are shown as sector plots (T: temporal, ST: superior-temporal, S: superior, SN: superior-nasal, N: nasal, NI: nasal-inferior, and I: inferior sector) with values for each group given as average \uf0b1 standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Non-sectorial parameters are presented as boxplots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' A significant difference between two groups (p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='05) was indicated with a * (determined by post-hoc Tukey HSD tests).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Critical points resulting from the three classification tasks: normal-mild, mild- moderate, and moderate advanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' From left to right column: 3D, en face (top), and sagittal (side) view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Surfaces represent the average anterior tissue boundaries for each respective dataset: RNFL+PLT (red), GCL+IPL (green), ORL (blue), RPE (yellow), choroid (purple), sclera (cyan), and LC (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Red colored critical points correspond to ONH regions with high importance for the differentiation of the respective glaucoma severity groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' 29 1 2 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' En face (top) view layer by layer comparison (columns) of critical points at different stages of glaucoma severity (rows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Critical points 3 are presented as point cloud density maps with colours indicating the number of neighbouring points within a sphere with a radius of 75 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' 4 30 Tables 5 6 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' Summary of glaucoma severity groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' 7 8 NORMAL (N=213) MILD (N=204) MODERATE (N=118) ADVANCED (N=118) P AGE, YEARS 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='36 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='99) 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='9 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='42) 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='05 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='11) 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='52 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='69) <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='001 SEX, FEMALE 126 (59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='15) 91 (44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='61) 49 (41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='52) 43 (36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='44) <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='001 RACE CHINESE 213 178 97 53 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='001 CAUCASIAN 0 26 21 65 MD, DB -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='41 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='11) -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='35 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='95) -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='16 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='35) -18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='64 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='31) <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content='001 Data are in mean (standard deviation) or n (%) as appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' 9 MD = mean deviation of the 24-2 or 30-2 visual field test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' 10 *Comparison between the four groups using Fisher’s exact test (for sex and race) and ANOVA 11 (for age and MD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} +page_content=' 12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE1T4oBgHgl3EQfAQKK/content/2301.02837v1.pdf'} diff --git a/49FKT4oBgHgl3EQf9y5B/content/tmp_files/2301.11955v1.pdf.txt b/49FKT4oBgHgl3EQf9y5B/content/tmp_files/2301.11955v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..33adf3ca691a00e7c0606bdab726cf36c0e0f5f1 --- /dev/null +++ b/49FKT4oBgHgl3EQf9y5B/content/tmp_files/2301.11955v1.pdf.txt @@ -0,0 +1,1414 @@ +Statistical whitening of neural populations with gain-modulating interneurons +Lyndon R. Duong * 1 David Lipshutz * 2 David J. Heeger 1 Dmitri B. Chklovskii 2 3 Eero P. Simoncelli 1 2 +Abstract +Statistical whitening transformations play a fun- +damental role in many computational systems, +and may also play an important role in biologi- +cal sensory systems. Individual neurons appear +to rapidly and reversibly alter their input-output +gains, approximately normalizing the variance of +their responses. Populations of neurons appear +to regulate their joint responses, reducing correla- +tions between neural activities. It is natural to see +whitening as the objective that guides these be- +haviors, but the mechanism for such joint changes +is unknown, and direct adjustment of synaptic in- +teractions would seem to be both too slow, and in- +sufficiently reversible. Motivated by the extensive +neuroscience literature on rapid gain modulation, +we propose a recurrent network architecture in +which joint whitening is achieved through modu- +lation of gains within the circuit. Specifically, we +derive an online statistical whitening algorithm +that regulates the joint second-order statistics of a +multi-dimensional input by adjusting the marginal +variances of an overcomplete set of interneuron +projections. The gains of these interneurons are +adjusted individually, using only local signals, +and feed back onto the primary neurons. The net- +work converges to a state in which the responses +of the primary neurons are whitened. We demon- +strate through simulations that the behavior of the +network is robust to poor conditioning or noise +when the gains are sign-constrained, and can be +generalized to achieve a form of local whitening +in convolutional populations, such as those found +throughout the visual or auditory system. +*Equal contribution 1Center for Neural Science, New York Uni- +versity, New York, NY 2Center for Computational Neuroscience, +Flatiron Institute, New York, NY 3Neuroscience Institute, New +York University School of Medicine, New York, NY. Correspon- +dence to: Lyndon R. Duong , David +Lipshutz . +Under review. +1. Introduction +Statistical whitening transformations, in which multi- +dimensional inputs are decorrelated and normalized to have +unit variance, are common in statistical signal processing +and machine learning systems. For example, they provide a +common step in statistical factorization methods (Hyv¨arinen +& Oja, 2000) and are often used as a preprocessing step +for training deep networks (Krizhevsky, 2009). Empiri- +cal evidence shows that statistical whitening improves un- +supervised feature learning (Coates et al., 2011). More +recently, self-supervised learning methods have used sta- +tistical whitening or related decorrelation transformations +to prevent representational collapse (Ermolov et al., 2021; +Zbontar et al., 2021; Bardes et al., 2021; Hua et al., 2021). +Whitening in neural networks is often performed in the of- +fline setting. However, online methods are useful, especially +when the inputs are from dynamic environments. +In early sensory systems, which receive inputs from dy- +namic environments, changes in sensory input statistics +induce rapid changes in the input-output gains of single +neurons, allowing cells to normalize their output variance +(Fairhall et al., 2001; Nagel & Doupe, 2006). This is hypoth- +esized to enable maximal information transmission (Barlow, +1961; Laughlin, 1981; Fairhall et al., 2001). At the popu- +lation level, whitening and related adaptive decorrelation +transformations have been reported in sensory areas such +as the early visual cortex of cats (Benucci et al., 2013) and +the olfactory bulb in zebrafish (Friedrich, 2013; Wanner & +Friedrich, 2020) and mice (Giridhar et al., 2011; Gschwend +et al., 2015). However, the mechanisms underlying such +whitening behaviors are unknown, and would seem to re- +quire coordination among all pairs of neurons, as opposed to +the single-neuron case which relies only on gain rescaling. +Here, motivated by the large neuroscience literature on rapid +gain modulation, we propose a novel recurrent network ar- +chitecture for statistical whitening that exclusively relies on +gain modulation. In particular, we introduce a novel objec- +tive for statistical whitening that is expressed solely in terms +of the marginal variances of an overcomplete representation +of the input signal. We derive a recurrent circuit to optimize +the objective, and show that it corresponds to a network +comprising primary neurons and an auxiliary population of +interneurons with scalar gain modulation. Importantly, the +arXiv:2301.11955v1 [q-bio.NC] 27 Jan 2023 + +Statistical whitening of neural populations with gain-modulating interneurons +Figure 1. Schematic of a recurrent statistical whitening network with 2 primary neurons and 3 interneurons. Left: 2D Scatter plot of the +(non-Gaussian) network inputs x = (x1, x2) whose covariance is the ellipse. Center: Primary neurons, whose outputs are y = (y1, y2), +receive external feedforward inputs, x, and recurrent feedback inputs from an auxiliary population of interneurons, − �3 +i=1 giziwi. +Linear projection vectors {w1, w2, w3} ∈ R2 encode non-negative feedforward synaptic weights connecting the primary neurons to +interneuron i = 1, 2, 3 (symmetric weights are used for feedback connections). The weights are shown in the left and right panels with +corresponding colors. Inset: The ith interneuron (e.g. here i = 2) receives input zi = w⊤ +i y, which is multiplied by its gain gi to produce +output gizi. Its gain, gi, is adjusted s.t. ∆gi ∝ z2 +i − 1. The dark arrow indicates that the gain update operates on a slower time scale. +Right: Scatter plots of the whitened network outputs y. Outputs have unit variance along all wi’s, which is equivalent to having identity +covariance matrix, i.e., Cyy = IN (black circle). +network operates online, and its responses converge to the +classical ZCA whitening solution without supervision or +backpropagation. To demonstrate potential applications of +this framework, we show that gain modulation serves as an +implicit gating mechanism, which facilitates fast context- +dependent whitening. Further, we show how non-negative +gain modulation provides a novel approach for dealing with +ill-conditioned or noisy data. Finally, we relax the overcom- +pleteness constraint in our objective and provide a method +for local decorrelation of convolutional populations. +2. A novel objective for ZCA whitening +Consider a neural network with N primary neurons. For +each t = 1, 2, . . . , let xt and yt be N-dimensional vectors +whose components respectively denote the inputs and out- +puts of the primary neurons at time t, Figure 1. Without loss +of generality we assume the inputs xt are centered. +2.1. Conventional objective +Statistical whitening aims to linearly transform inputs xt so +that the covariance of the outputs yt is identity, i.e., +Cyy = ⟨yty⊤ +t ⟩t = IN, +(1) +where ⟨·⟩t denotes the expectation operator over t, and IN +denotes the N × N identity matrix (see Appendix A for a +list of notation used in this work). +It is well known that whitening is not unique: any orthog- +onal rotation of a random vector with identity covariance +matrix also has identity covariance matrix. There are sev- +eral common choices to resolve this rotational ambiguity, +each with their own advantages (Kessy et al., 2018). Here, +we focus on the popular whitening transformation called +Zero-phase Component Analysis (ZCA) whitening or Ma- +halanobis whitening, which is the whitening transformation +that minimizes the mean-squared error between the inputs +and the whitened outputs (alternatively, the one whose trans- +formation matrix is symmetric). Mathematically, the ZCA- +whitened outputs are the optimal solution to the minimiza- +tion problem +min +{yt}⟨∥xt − yt∥2 +2⟩t +s.t. +⟨yty⊤ +t ⟩t = IN, +(2) +where ∥ · ∥2 denotes the Euclidean norm on RN. Assuming +the covariance of the inputs Cxx := ⟨xtx⊤ +t ⟩t is positive +definite, the unique solution to the optimization problem +in Equation 2 is yt = C−1/2 +xx +xt for t = 1, 2, . . . , where +C−1/2 +xx +is the inverse matrix square root of Cxx. +Equation 2 provides a starting point for deriving online ZCA +whitening algorithms that can be implemented with recur- +rent neural networks that learn by updating their synaptic +weights (Pehlevan & Chklovskii, 2015). +2.2. A novel objective using marginal statistics +We formulate a novel objective for learning the ZCA whiten- +ing transform via gain modulation. Our innovation exploits +the fact that a random vector has identity covariance matrix +(i.e., Equation 1 holds) if and only if it has unit marginal + +g12 +W2,1 +9222 +1 +1 +W: +92之。 + W2,2 +Close-up of + gain-modulated interneuron +Input:(1, 2Statistical whitening of neural populations with gain-modulating interneurons +variance along all possible 1D projections (a form of to- +mography; see Related Work). We can derive a tighter +statement, that holds for a finite but overcomplete set of at +least K ≥ KN := N(N + 1)/2 distinct axes (‘overcom- +plete’ simply means that the number of axes exceeds the +dimensionality of the input, i.e., K > N). Intuitively, this +equivalence holds because an N × N symmetric matrix has +KN degrees of freedom, so the marginal variances along +K ≥ KN distinct axes are sufficient to constrain the N ×N +(symmetric) covariance matrix. We formalize this equiva- +lence in the following proposition, whose proof is provided +in Appendix B. +Proposition 2.1. Fix K ≥ KN. Suppose w1, . . . , wK ∈ +RN are unit vectors1 such that +span({w1w⊤ +1 , . . . , wKw⊤ +K}) = SN, +(3) +where SN denotes the KN-dimensional vector space of N × +N symmetric matrices. Then Equation 1 holds if and only if +the projection of yt onto each unit vector w1, . . . , wK has +unit variance, i.e., +⟨(w⊤ +i yt)2⟩t = 1 +for +i = 1, . . . , K. +(4) +Assuming Equation 3 holds, we can interpret the set of +vectors {w1, . . . , wK} as a frame (i.e., an overcomplete +basis; Casazza et al., 2013) in RN such that the covariance +of the outputs Cyy can be computed from the variances of +the K-dimensional projection onto the set of frame vectors. +Thus, we can replace the whitening constraint in Equation 2 +with the equivalent marginal variance constraint to obtain +the following objective: +min +{yt}⟨∥xt − yt∥2 +2⟩t +s.t. +Equation 4 holds. +(5) +3. A recurrent neural network with gain +adaptation for ZCA whitening +In this section, we derive an online algorithm for solving +the optimization problem in Equation 5 and map the algo- +rithm onto a recurrent neural network with gain modulation. +We first introduce Lagrange multipliers to enforce the con- +straints, which transforms the minimization problem into a +minimax problem. We then solve the minimax problem by +taking stochastic gradient steps. +Assume we have an overcomplete frame {w1, . . . , wK} in +RN satisfying Equation 3. We concatenate the frame vectors +into an N ×K matrix W := [w1, . . . , wK]. In our network, +primary neurons project onto the layer of K interneurons +with the synaptic weights representing matrix W. Then, +the post-synaptic currents in interneurons at time t encode +1The unit-length assumption is without loss of generality and +is imposed here for notational convenience. +the K-dimensional vector zt := W⊤yt (Figure 1). We +emphasize that the synaptic weight matrix W will remain +fixed in our whitening algorithm. +3.1. Enforcing the marginal variance constraints with +scalar gains +We introduce Lagrange multipliers g1, . . . , gK ∈ R to en- +force the K constraints in Equation 4. We concatenate +the Lagrange multipliers into the K-dimensional vector +g := [g1, . . . , gK]⊤ ∈ RK, and formulate the problem as a +saddle point optimization, +max +g +min +{yt}⟨ℓ(xt, yt, g)⟩t, +(6) +where ℓ(x, y, g) := ∥x − y∥2 +2 + +K +� +i=1 +gi +� +(w⊤ +i y)2 − 1 +� +. +Here, we have interchanged the order of maximization over +g and minimization over yt, which is justified because +ℓ(xt, yt, g) is convex in yt and linear in g, see Appendix C. +In our neural network implementation, gi will correspond +to the multiplicative gain associated with the ith interneuron, +so that its output at time t is gizi,t (Figure 1, Inset). From +Equation 6, we see that the gain of the ith interneuron, gi, +enforces the marginal variance of yt along the axis spanned +by wi to be unity. Importantly, the gains are not hyper- +parameters, but rather they are optimization variables which +promote statistical whitening of {yt}, preventing the neural +outputs from trivially matching the inputs {xt}. +3.2. Deriving recurrent neural network update rules +To solve Equation 6 in the online setting, we assume there +is a time-scale separation between ‘fast’ neural dynamics +and ‘slow’ gain updates, so that at each time step the neural +dynamics equilibrate before the gains are adjusted. This al- +lows us to perform the inner minimization over {yt} before +the outer maximization over the gains. In biological neural +networks, this is justifiable because a given neuron’s activa- +tions (i.e. action potential firing) operate on a much more +rapid time-scale than its intrinsic input-output gain, which +is driven by slower processes such as changes in calcium +ion concentration gradients (Ferguson & Cardin, 2020). +3.2.1. FAST NEURAL ACTIVITY DYNAMICS +For each time step t = 1, 2, . . . , we minimize the objective +ℓ(xt, yt, g) over yt by recursively running gradient-descent +steps to equilibrium: +yt ← yt − γ +2 ∇yℓ(xt, yt(τ), g) += yt + γ {xt − W(g ◦ zt) − yt} , +(7) +where γ > 0 is a small constant, the circle ‘◦’ denotes the +Hadamard (element-wise) product, g ◦ zt is a vector of K + +Statistical whitening of neural populations with gain-modulating interneurons +gain-modulated interneuron outputs, and we assume the +primary cell outputs are initialized at zero. +We see from the right-hand-side of Equation 7 that the ‘fast’ +dynamics of the primary neurons are driven by three terms +(inside the curly braces): i) constant feedforward external +input xt; ii) recurrent gain-modulated feedback from in- +terneurons −W(g ◦ zt); and iii) a leak term −yt. Because +the neural activity dynamics are linear, we can analytically +solve for their equilibrium (i.e. steady-state), ¯yt, by setting +the update in Equation 7 to zero: +¯yt = +� +IN + W diag (g) W⊤�−1 xt += +� +IN + +K +� +i=1 +giwiw⊤ +i +�−1 +xt, +(8) +where diag (g) denotes the K × K diagonal matrix whose +(i, i)th entry is gi, for i = 1, . . . , K. The equilibrium feed- +forward interneuron inputs are then given by +¯zt = W⊤¯yt. +(9) +The gain-modulated outputs of the K interneurons, g ◦ zt, +are then projected back onto the primary cells via symmetric +weights, −W (Figure 1). +3.2.2. SLOW GAIN DYNAMICS +After the fast neural activities reach steady-state, the in- +terneuron gains are updated by taking a stochastic gradient- +ascent step with respect to g: +g ← g + η +2∇gℓ(xt, ¯yt, g) += g + η +�¯z◦2 +t − 1 +� +, +(10) +where η +> +0 is the learning rate, the superscript +‘◦2’ denotes the element-wise squaring operation (i.e., +¯z◦2 +t += [¯z2 +t,1, . . . , ¯z2 +t,K]⊤) and 1 = [1, . . . , 1]⊤ is the K- +dimensional vector of ones2. Remarkably, the update to the +ith interneuron’s gain gi (Equation 10) depends only on the +online estimate of the variance of its equilibrium input ¯z2 +t,i, +and its distance away from the target variance, 1. Networks +such as these which adapt using only local signals to each +interneuron are suitable candidates for hardware implemen- +tations using low-power neuromorphic chips (Pehlevan & +Chklovskii, 2019). Thus, although statistical whitening in- +herently requires a joint transformation in response to joint +statistics, our recurrent network solution operates solely +using single-neuron gain changes in response to marginal +statistics. +2Appendix D generalizes the gain update to allowing for +temporal-weighted averaging of the variance over past samples. +3.2.3. ONLINE UNSUPERVISED ALGORITHM +By combining Equations 7 – 10, we arrive at our online +recurrent neural network algorithm for statistical whitening +via gain modulation (Algorithm 1). We also provide batched +and offline versions of the algorithm in Appendix E. +Algorithm 1 Online ZCA whitening via gain modulation +1: Input: Centered inputs x1, x2, · · · ∈ RN +2: Initialize: W ∈ RN×K; g ∈ RK; η, γ > 0 +3: for t = 1, 2, . . . do +4: +yt ← 0 +5: +{Run yt and zt dynamics to equilibrium} +6: +while not converged do +7: +zt ← W⊤yt +8: +yt ← yt + γ {xt − W(g ◦ zt) − yt} +9: +end while +10: +g ← g + η +� +z◦2 +t − 1 +� +{Update gains} +11: end for +There are a few points worth noting about this network: +• The weights W remain fixed in Algorithm 1. Rather, +the gains g adapt to statistically whiten the outputs. +This allows the whitening to be easily adjusted and +reversed, by simply returning the gains to their default +states. +• While the objective is effectively in the form of an auto- +encoding loss function involving an ℓ2 reconstruction +term (Eq. 6), the recurrent network never explicitly +reconstructs its inputs. +• Since all recurrent dynamics are linear, it is possible to +bypass the inner loop representing the fast dynamics +of the primary cells (lines 6 – 9 of Algorithm 1), by +directly computing the equilibrium responses of ¯yt, +and ¯z directly (Eqs. 8, 9). +4. Numerical experiments and applications +We provide different applications of our recurrent ZCA +whitening network via gain modulation. In particular, we +emphasize that gain adaptation is distinct from, while also +complementary to, a synaptic weight learning. We therefore +side-step the goal of learning the frame W, and assume it +is known. This allows us to decouple and analyze the gen- +eral properties of our proposed gain modulation framework, +independently from the choice of frame. +4.1. Gain modulation: a new solution to ZCA +whitening +We first demonstrate that our algorithm succeeds in yielding +statistically whitened outputs. We simulated a network with +interneuron weights, W, as illustrated in Figure 1 (N=2, + +Statistical whitening of neural populations with gain-modulating interneurons +Figure 2. Network from Figure 1 (with corresponding colors; +N=2, K=KN=3, η=2E-3) whitening to two randomly gener- +ated statistical contexts online (10K steps each). Top: Marginal +variances (log scale) measured by interneurons approach 1 over +time. Middle: Dynamics of interneuron gains, which are applied +to zi before feeding back onto the primary cells. Dashed lines are +optimal gains (Appendix F). Bottom: Whitening error over time. +K=KN=3). Figure 2 shows network adaptation to inputs +from two contexts with randomly generated underlying in- +put covariances Cxx (10K gain update steps each). As up- +date steps progress, all marginal variances converge to unity, +as expected from the objective (top panel). To achieve ZCA +whitening at equilibrium, then IN +�K +i=1 giwiw⊤ +i = C1/2 +xx +(Equation 8). When the number of interneurons satisfies +K=KN, the optimal gains to achieve ZCA whitening can +be solved analytically (see Appendix F for details). These +are displayed as dashed lines in the (middle panel). We +found that the network successfully adapted to the two ran- +dom statistical contexts, and converged to the optimal set +of gains to achieve whitened yt (Figure 2). Accordingly, +the whitening error, as measured by the Frobenius norm be- +tween Cyy and IN, approached zero (bottom panel). Thus, +with each interneuron monitoring their respective marginal +input variances z2 +i , and re-scaling their input-output gains to +modulate feedback onto the primary neurons, the network +succeeded in adapting to each context and yielded whitened +outputs. +4.2. Rate of convergence depends on frame W +Thus far, we have assumed the frame, W, was fixed and +known (e.g., optimized through pre-training or long time- +scale development). This distinguishes our method from +existing ZCA whitening methods, which typically operate +by estimating the eigenvectors of the data. By contrast, our +network obviates learning the principal axes of the data +altogether, and instead uses a statistical sampling approach +along a fixed set of measurement axes. +If the number of interneurons K=KN, their gains will de- +scend the gradient of the objective (Equation 10), and by +Proposition Theorem 2.1, the outputs will become whitened. +We were interested in how effectively the network whitened +randomly sampled inputs with fixed input covariance de- +pending on its initialization. Figure 3 summarizes an empir- +ical convergence test of 100 networks where N = 2 with +three different kinds of frame W ∈ RN×KN : i) with i.i.d. +Gaussian entries (‘Random’); ii) through an optimization +procedure that finds a frame whose columns have mini- +mum mutual coherence and cover the ambient space (‘Opti- +mized’); and iii) a frame whose first N columns were the +eigenvectors of the data and the remaining KN −N columns +were random Gaussian entries (‘Spectral’). For clarity, we +have removed the effects of sampling stochasticity by run- +ning the offline version of our network, which assumes +having direct access to the input covariance (Appendix E); +the online version was qualitatively similar. +Figure 3. Convergence depends on qualitative structure of W. +Networks each had N=2, K=KN=3, η=5E-3. Shaded error +regions are standard errors over the 100 repeats. +The Spectral frame defines a bound on achievable perfor- +mance, converging much faster than the Random and Op- +timized frames. This is because the interneuron axes were +aligned with the input’s principal axes, and a simple gain +scaling along those directions is the optimal whitening solu- +tion. Interestingly, we found that networks with optimized +weights systematically converged faster than randomly- +initialized frames. These results indicate that the choice +of frame does in fact play an important role in the effective- +ness of our algorithm. Namely, increased coverage of the +space by the frame vectors facilitates whitening with our +gain re-scaling mechanism. The random sampling approach +has little hope of scaling to high dimensional inputs, and the +green line in Figure 3 shows that one would benefit from +aligning the frame vectors to the principal axes of the inputs. +4.3. Implicit gating via gain modulation +Motivated by the findings in Figure 3, we wished to demon- +strate a way in which our adaptive gain modulation net- +work could complement or augment a network in which +context-dependent weights have already been learned. We +performed an experiment involving a network with ‘pre- +trained’ W (N=6, K=KN=21) whitening inputs from + +100 +2 +ICy-Inl +2F +10-2 +10-6 +0 +10000 +20000 +Stepw +Random +Optimized +10-1 +Spectral +10-2 +10-3 +0 +200 +400 +600 +800 +1000 +StepStatistical whitening of neural populations with gain-modulating interneurons +Figure 4. Gains can act as an implicit gating mechanism. Top: +Whitening error over time with a network (N=6; KN=21; η=1E- +3) adapting to 2 alternating statistical contexts A and B, with +different input covariances for 10K steps each. W was initialized +as a Spectral frame, with the first 2N columns set to be the eigen- +vectors of covariances of contexts A and B, respectively. Bottom: +Gains can be seen to act as switches for context, gating the spectral +components to optimally whiten each context. +two alternating statistical contexts, A and B, for 10K steps +each. The frame was constructed such that the first and +second N columns were the eigenvectors of context A and +B’s covariance, respectively, and the remaining K − 2N +columns’ elements were random i.i.d. Gaussian. Figure 4 +(top panel) shows that the network adaptively whitens the +inputs from each successive context. Surprisingly, upon +closer inspection to the K interneurons’ gains over time +(bottom panel) showed that they approximately served to +‘select’ the frame vectors corresponding to the eigenvectors +of each respective condition (as indicated by the blue/red in- +tensity on the figure). Our gain modulation framework thus +serves as an effective means of gating context-dependent +information without an explicit context signal. +4.4. Normalizing ill-conditioned data +When inputs are low-rank, Cxx is ill-conditioned (Fig- +ure 5A), and whitening can amplify directions of small +variance that are due to noise. In this section, we show how +our gain-modulating network can be simply modified to han- +dle these types of inputs. To prevent amplification of inputs +below a certain threshold, we can replace the unit marginal +variance equality constraints with upper bound constraints: +⟨(w⊤ +i yt)2⟩t ≤ 1 +for +i = 1, . . . , K. +(11) +Our modified network objective then becomes +min +{yt}⟨∥xt − yt∥2 +2⟩t +s.t. +Equation 11 holds. +(12) +Figure 5. Two networks (N=2, K=3, η=0.02) whitening ill- +conditioned inputs. A: Outputs without whitening. 2D scatterplot +of a non-Gaussian density whose underlying signal lies close to +a latent 1D axis. The signal magnitude along that axis is denoted +by the colors. The covariance matrix is depicted as a black ellipse. +Gray dashed lines are axes spanned by W (here chosen to be an +equi-angular frame). B: ZCA whitening boosts small-amplitude +noise lying along the uninformative direction. C: Modulating gains +according to Eq. 14 rescales the data without amplifying noise. D: +Gains updated with Eq. 10 (solid) vs. Eq. 14 (dashed). +Intuitively, if the projected variance along a given direction +is already less than or equal to unity, then it will not affect +the overall loss. To enforce the upper bound constraints, +we introduce gains as Lagrange multipliers as before, but +restrict the domain of g to be the non-negative orthant RK ++ , +resulting in non-negative optimal gains: +max +g∈RK ++ +min +{yt}⟨ℓ(xt, yt, g)⟩t, +(13) +where ℓ(x, y, g) is defined as in Equation 6. At each time +step t, we optimize Equation 13 by first taking gradient- +descent steps with respect to yt, resulting in the same neu- +ral dynamics (Equation 7) and equilibrium solution (Equa- +tion 8) as before. After the neural activities equilibrate, we +take a projected gradient-ascent step with respect to g: +g ← ⌊g + η(¯z◦2 +t − 1)⌋ +(14) +where ⌊·⌋ denotes the element-wise half-wave rectification +operation that projects its inputs onto the positive orthant +RK ++ , i.e., ⌊v⌋ := [max(v1, 0), . . . , max(vK, 0)]⊤. +We simulated a network with gains set to either updates +using unconstrained gains (Equation 10), or rectified gains +(Equation 14), and observed that these two models con- +verged to two different solutions (Figure 5B, C). When + +10-2 +Context A +Context B +Context A +Context B +10-3 +10-4 +10-5 +0 +Interneuron index +5 +10 +15 +20 +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +Step (x10) +-0.20 +-0.15 +-0.10 +-0.05 +0.00 +0.05 +0.10 +0.15 +0.20 +GainA +B +2 +2 +1 +1 +0 +0 +-1 - +-1 +-2 +-2 +1 +T +- +-2 +-1 +0 +1 +2 +-2 +-1 +0 +1 +2 +y1 +y1 +C +D +0.75 +2 +0.50 +0.25 +1 +0.00 +0 +9 +-0.25 +-0.50 +-1 +-0.75 +gi +-2 +-1.00 +Lgi] +-2 +-1 +0 +1 +2 +0 +100 +200 +300 +y1 +StepStatistical whitening of neural populations with gain-modulating interneurons +gi was not constrained to be non-negative, the network +achieved global whitening, as before. By contrast, the gains +constrained to be non-negative converged to different val- +ues altogether, with one of them converging to zero rather +than becoming negative. The whitening error for this net- +work unsurprisingly converged to a non-zero value with the +non-negative gain constraint. Thus, with a non-negative +constraint, the network failed to fully whiten y, but in doing +so, it did not amplify the noise. In Appendix G we show +additional cases that provide further geometric intuition on +differences between ZCA whitening and non-negative gain +constrained ZCA whitening with our network. +4.5. Gain modulation enables local spatial +decorrelation +The requirement of KN interneurons to ensure a statisti- +cally white output becomes prohibitively costly for high- +dimensional inputs due to the number of interneurons scal- +ing as O(N 2). This led us to ask: how many interneurons +are needed in practice? For natural sensory inputs such +as images, it is well known that inter-pixel correlation is +highly structured, decaying as a function of distance. Using +a Gaussian random walk, we simulated gaze fixation and +micro-saccadic eye movements, drawing 12×12 patch sam- +ples from a natural image (Figure 6A; Hateren & Schaaf, +1998). We did this for different randomly selected regions +of the image (colors). The content of each region is quite +different, but the inter-pixel correlation within each context +fell rapidly with distance (Figure 6B). +We relaxed the O(N 2) marginal variance constraint to in- +stead target whitening of spatially local neighborhoods of +primary neurons with image patch inputs. That is, the frame +W spanned K < KN axes in RN, but was constructed such +that overlapping neighborhoods of 4 × 4 primary neurons +were decorrelated, each by a population of interneurons that +was ‘overcomplete’ with respect to that neighborhood (see +Appendix H for frame construction details). Importantly, +taking into account convolutional structure dramatically re- +duces the interneuron complexity from O(N 2) → O(N) +(Appendix H). This frame is still overcomplete (K > N), +but because K +90 +91 +0.07 +0.09 +0.11 +r0=103M, M=103MBH +r0=102M, M=103MBH +r0=103M, M=102MBH +r0=102M, M=102MBH +M=0 +30 +40 +50 +60 +70 +80 +90 +100 +10 +20 +30 +40 +50 +p/Rs +P/hour +90 +91 +41 +41.5 +FIG. 2. +The results of orbital period and precession for EMRIs in galaxies with and without DM. +The mass of central MBHs is set as MBH = 106M⊙ and the eccentricity e = 0.6. We take the +compactness M/r0 as 10−2 and 10−3, and the total mass M as 102MBH, 103MBH and M = 0. The +inserts show the evolution in a short time period. +III. +GWS OF EMRIS IN THE ENVIRONMENTS OF GALAXIES +Using the above results for the orbital motions of EMRIs, we get the leading order energy +and angular momentum fluxes +�dE +dt +� +GW +≃ 32 +5 +� +µ +MBH +�2 �MBH +p +�5 +(1 − e2)3/2 +� +1 + 73 +24e2 + 37 +96e4 +� � +1 − 6M +r0 +� +, +(22) +�dL +dt +� +GW +≃ 32 +5 +� +µ +MBH +�2 +MBH +�MBH +p +�7/2 +(1 − e2)3/2 +� +1 + 7 +8e2 +� � +1 − 5M +r0 +� +. +(23) +The last factors 1 − 6M/r0 and 1 − 5M/r0 are the corrections from DM halos around the +MBH. Note that the effects of environmental DM halos on the losses of energy and angular +momentum only depend on the compactness M/r0 and the energy and angular momentum +fluxes become smaller if the compactness is larger. In the absence of local DM halos, M = 0, +Eqs. (22) and (23) recover the standard results for eccentric binaries [63, 64]. Applying the +energy and angular momentum balance equations +�dE +dt +� +GW += − +�dE +dt +� +orbit +, +(24) +�dL +dt +� +GW += − +�dL +dt +� +orbit +, +(25) + +10 +we get the leading order evolution of the orbital parameters p(t) and e(t) due to the emission +of GWs, +dp +dt = −64 +5 +µ +MBH +�MBH +p +�3 � +1 − e2� 3 +2 +� +1 + 7 +8e2 +� � +1 − 5M +r0 +� +, +(26) +de +dt = −304 +15 +e +p +µ +MBH +�MBH +p +�3 � +1 − e2� 3 +2 +� +1 + 121 +304e2 +� � +1 − 5M +r0 +� +. +(27) +Since the right sides of Eqs. (26) and (27) are negative, both the semi-latus rectum p and +the eccentricity decrease with time due to the radiation of GWs. The presence of local DM +halos slows down the decrease of p and e, the bigger the compactness M/r0 is, the slower +the semi-latus rectum p(t) and the eccentricity decrease. In Fig. 3, we show the evolution +of the orbital parameters p(t) and e(t) due to the emission of GWs. Comparing with the +astrophysical environments without DM, it takes more time for EMRIs with DM halos to +evolve from p = 20Rs to p = 3Rs. The larger the compactness M/r0 is, the more time it +takes. The presence of DM halos also slows down the decrease rate of the eccentricity and +the final eccentricity is a bit larger with larger compactness. +r0=102M, e0=0.6 +r0=103M, e0=0.6 +r0=102M, e0=0.2 +r0=103M, e0=0.2 +M=0, e0=0.6 +M=0, e0=0.2 +0 +200 +400 +600 +800 +1000 +0 +5 +10 +15 +20 +t/yr +p/Rs +0 +200 +400 +600 +800 +1000 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +t/yr +e +FIG. 3. +The evolution of the orbital parameters p and e from the initial p = 20Rs to p = (3+e)Rs. +The mass of central MBHs is chosen as MBH = 106M⊙, the mass of the SCO is µ = 10M⊙ and the +initial eccentricity is chosen as e0 = 0.2, 0.6. We consider two different values for the compactness +of the DM halo, M/r0 = 10−2 and 10−3. The solid lines correspond to the cases without DM. +As discussed above, the effects of DM halos will be manifested in GW waveforms. The +quadrupole formula of GWs is +hjk = 2 +dL +¨Ijk, +(28) + +11 +where dL is the luminosity distance between the detector and the source and Ijk is the +quadrupole moment of EMRIs. The tenser modes h+ and h× in the transverse-traceless +gauge are given by +h+ = 1 +2 +� +ej +Xek +X − ej +Y ek +Y +� +hjk, +(29) +h× = 1 +2 +� +ej +Xek +Y − ej +Y ek +X +� +hjk, +(30) +where eX and eY are the orthonormal vectors in the plane that is perpendicular to the +direction from the detector to the GW source. Plugging the results for the orbital evolution +obtained above into Eq. (28), we numerically calculate the time-domain GW waveforms. +The time-domain plus-mode GW waveforms for EMRIs with and without DM halos are +shown in Fig. 4. From Fig 4, we see that initially the difference between GW waveforms +with and without DM halos is negligible. One year later, the two waveforms for EMRIs with +and without DM halos are quite different. +In order to quantify the impact of DM halo environments on the dephasing of GW +waveforms, we calculate the number of orbital cycles accumulated from time ti to tf [65–67] +N(t) = +� tf +ti +˙φ(t)dt. +(31) +Over one-year evolution before the merger, the numbers of orbital cycles for EMRIs with +and without DM halos are NDM and N0 respectively. In Fig 5, we show the difference ∆N = +NDM − N0 between the number of orbital cycles with and without DM halos accumulated +over one year before the merger. Following [68], we choose ∆N ∼ 1 rad as the threshold for +a detectable dephasing. The results show that we can detect the compactness as small as +≲ 10−4. The results also show that eccentric orbits can help detect DM halos with smaller +compactness. +To distinguish the waveforms more accurately, we calculate the mismatch between GW +signals emitted from EMRIs with and without DM halos. Given two signals h1(t) and h2(t), +the inner product (h1|h2) is defined as +(h1|h2) = 2 +� +∞ +0 +˜h1(f)˜h∗ +2(f) + ˜h2(f)˜h∗ +1(f) +Sh(f) +df, +(32) +where ˜h(f) is the Fourier transformation of the time-domain signal h(t), ˜h∗ denotes the +complex conjugate of ˜h, and the SNR for the signal h is +� +(h|h). For LISA, the one-side + +12 +0 +20 +40 +60 +80 +100 +-5 +0 +5 +t/hour +h+ +At the beginning +0 +20 +40 +60 +80 +100 +-5 +0 +5 +t/hour +h+ +365 days later +r0=102M, M=102MBH +M=0 +1023× +1023× +0 +20 +40 +60 +80 +100 +-5 +0 +5 +t/hour +h+ +0 +20 +40 +60 +80 +100 +-5 +0 +5 +t/hour +h+ +r0=103M, M=102MBH +M=0 +1023× +1023× +FIG. 4. +The time-domain plus mode GW waveforms for EMRIs with and without DM halos. +The mass of central MBHs is MBH = 106M⊙, the mass of the SCO is µ = 10M⊙, the total mass +of DM halos M is = 102MBH, the inclination angle ι = π/6, the luminosity distance dL = 1Gpc, +the initial longitude of pericenter ω0 = 0 and the initial eccentricity e0 = 0.6 at p0 = 20Rs. M = 0 +corresponds to the case without DM halos. The left panels show the initial waveforms. The right +panels show the waveforms after one year. The top panels are for M/r0 = 10−2 and the bottom +panels are for M/r0 = 10−3. +noise power spectral density is [69] +Sh(f) = Sx +L2 + 2Sa [1 + cos2(2π fL/c)] +(2 πf)4L2 +� +1 + +�4 × 10−4Hz +f +�� +, +(33) +where √Sa = 3 × 10−15 m s−2/Hz1/2 is the acceleration noise, √Sx = 1.5 × 10−11 m/Hz1/2 +is the displacement noise and L = 2.5 × 106 km is the arm length of LISA [7]. The overlap +between two GW signals is quantified as [60] +O(˜h1, ˜h2) = +(˜h1|˜h2) +� +(˜h1|˜h1)(˜h2|˜h2) +, +(34) + + ro=102M, M=102MBH +- M=0ro=103M, M=102MBH +M-013 +e0=0 +e0=0.2 +e0=0.4 +e0=0.6 +-5 +-4.5 +-4 +-3.5 +-3 +-2.5 +-2 +300 +250 +200 +150 +100 +50 +0 +Log10[M/r0] +|Δ| +FIG. 5. +The difference between the orbital cycles with and without DM halos ∆N(t) over one-year +evolution before the merger for different compactness of halos M/r0. The initial eccentricity e0 +is chosen at p0 = 20Rs. The mass of central MBHs is MBH = 106M⊙ and the mass of the SCO +is µ = 10M⊙. The masses of DM halos are M = 102MBH. The black dashed line corresponds to +∆N = 1 rad. +and the mismatch between two signals is defined as +Mismatch = 1 − Omax(˜h1, ˜h2), +(35) +where the maximum is evaluated with respect to time and phase shifts. The mismatch is +zero if two signals are identical. Two signals are considered experimentally distinguishable if +their mismatch is larger than d/(2 SNR2), where d = 13 is the number of intrinsic parameters +of the GW source [70–72]. Considering EMRIs with masses (106+10)M⊙ at dL = 1 Gpc and +integration time of one year before the coalescence, we calculate the mismatch between GW +waveforms with and without DM halos and the results with LISA are shown in Fig 6. The +SNR is about 32 for the GW signals from EMRIs considered above. The initial eccentricity +e0 is chosen at p0 = 20Rs. As shown in Fig 6, if the compactness of DM halo M/r0 is +larger, then the mismatch between GW waveforms with and without DM halos is bigger, +so more compact DM halos can be detected easier with LISA. Again eccentric orbits can +detect smaller compactness. Therefore, we can use GWs from EMRIs in the environments +of galaxies to test the existence of DM halos and detect the compactness of the halos M/r0 +as small as 10−5. + +14 +e0=0.2 +e0=0.6 +-6 +-5 +-4 +-3 +-2 +-1 +0.001 +0.010 +0.100 +1 +Log10[M/r0] +Mismatch +FIG. 6. +The results of the mismatch between GW waveforms with and without DM halos for +different compactness M/r0 and initial eccentricity e0. The black dashed line corresponds to the +threshold d/(2 SNR2) ≈ 0.0072. +IV. +CONCLUSIONS AND DISCUSSIONS +Using the analytic, static and spherically symmetric metric for a Schwarzschild black hole +immersed in DM halos with Hernquist type density distribution, we derive analytic formulae +for the orbital period and orbital precession for eccentric EMRIs with the environment of +DM halos. The results show that the presence of DM halo decreases the orbital procession +and even retrogrades the orbital procession if the local density of DM halos ρDM ∼ M/r2 +0 is +large enough. As the orbit becomes larger, the orbital precession decreases and the prograde +precession decreases faster in the presence of DM halos. With DM halos, the prograde-to- +retrograde precession transition happens at some critial value of p and then the prograde +precessions change to retrograde precessions as p increases further; afterwards, the retrograde +precessions increase as p increases. +Taking the energy and angular momentum fluxes of GWs into consideration, we derive +analytic formulae for the evolutions of the semi-latus rectum and the eccentricity. +The +presence of local DM halos slows down the decrease of the semi-latus rectum and the eccen- +tricity. Comparing the numbers of orbital cycles with and without DM halos over one-year +evolution before the merger, we find that DM halos with the compactness as small as 10−4 +can be detected. By calculating the mismatch between GW waveforms with and without +DM halos, we show that we can use GWs from EMRIs in the environments of galaxies to + +15 +test the existence of DM halos and detect the compactness as small as 10−5. 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Maselli, Gravitational waves in massive gravity theories: wave- +forms, fluxes and constraints from extreme-mass-ratio mergers, Phys. Rev. Lett. 121, 251103 +(2018), arXiv:1809.00673 [gr-qc]. + diff --git a/8NE4T4oBgHgl3EQfdAy1/content/tmp_files/load_file.txt b/8NE4T4oBgHgl3EQfdAy1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c151d0701c4fa9d20289bb8216526ce36d62fcec --- /dev/null +++ b/8NE4T4oBgHgl3EQfdAy1/content/tmp_files/load_file.txt @@ -0,0 +1,936 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf,len=935 +page_content='Extreme mass ratio inspirals in galaxies with dark matter halos Ning Dai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' ∗ Yungui Gong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' † Yang Zhao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' ‡ and Tong Jiang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' § 1School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Huazhong University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' 1037 LuoYu Rd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Wuhan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Hubei 430074,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' China Using the analytic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' static and spherically symmetric metric for a Schwarzschild black hole immersed in dark matter (DM) halos with Hernquist type density distri- bution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' we derive analytic formulae for the orbital period and orbital precession,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' the evolutions of the semi-latus rectum and the eccentricity for eccentric EMRIs with the environment of DM halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' We show how orbital precessions are decreased and even reverse the direction if the density of DM halo is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The presence of local DM halos slows down the decrease of the semi-latus rectum and the eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Comparing the number of orbital cycles with and without DM halos over one-year evolution before the merger, we find that DM halos with the compactness as small as 10−4 can be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' By calculating the mismatch between GW waveforms with and without DM halos, we show that we can use GWs from EMRIs in the environ- ments of galaxies to test the existence of DM halos and detect the compactness as small as 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' INTRODUCTION The first detection of gravitational waves (GWs) from the merger of black hole (BH) binary by the LIGO Scientific Collaboration and the Virgo Collaboration in 2015 [1, 2] opened a new window for probing gravitational physics and fundamental physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Since then, tens of confirmed GW events have been detected by the ground-based GW observatories [3– 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The ground-based GW observatories are only sensitive to GWs in the frequency range of 10 − 103 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The space-based GW observatories such as LISA [7], TianQin [8] and Taiji [9, 10] will usher a new era in GW astronomy due to their unprecedented accuracy and their sensitive range of mHz [11–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' One particular interesting target of space-based ∗ daining@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='cn † Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' yggong@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='cn ‡ zhaoyangedu@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='cn § jiangtong@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='cn arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='05088v1 [gr-qc] 12 Jan 2023 2 GW detectors is a stellar-mass compact object (SCO) inspiralling onto a massive black hole (MBH), the extreme mass ratio inspirals (EMRIs) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' There are 105 − 106 GW cycles in the detector band when the SCO inspirals deep inside the strong field region of the MBH, and rich information about the spacetime geometry around the MBH is encoded in GW waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Therefore, the observations of GWs emitted from EMRIs present us a good opportunity for the study of astrophysics, gravity in the strong and nonlinear regions and the nature of BHs [15–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Although the property of DM is still a mystery in physics, there are a lot of indirect evidence for the existence of dark matter (DM) in the Universe [21–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' DM may cluster at the center of galaxies and around BHs [31–34], and affect the dynamics of binaries and hence GWs emitted from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Since EMRIs are believed to reside in stellar clusters and the center of galaxies, so DM may affect the dynamics of EMRIs and the observations of GWs from EMRIs, especially those in DM environments may be used to understand the astrophysical environment surrounding EMRIs and probably confirm the existence of DM and uncover the nature of DM [35–49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' In the studies of DM effects discussed above, Newtonian approaches to the problems were applied and the gravitational effects of DM on the dynamical evolution of EMRIs were mod- eled at Newtonian level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' [50], the authors generalized Einstein clusters [51, 52] to include horizons, solved Einstein’s equations sourced by DM halo of Hernquist type density distribution [34] with a MBH at its center and obtained analytical formulae for the metric of galaxies harboring MBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Exact solutions for the geometry of a MBH immersed in DM halos with different density distributions were then derived [53, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' With the fully relativis- tic formalism, it was found that the leading order correction to the ringdown stage induced by the external matter and fluxes by orbiting particles is a gravitational redshift, and the difference between the number of GW cycles accumulated by EMRIs with and without DM halos over one year before the innermost stable circular orbit can reach about 500 [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' In galaxies harboring MBHs, tidal forces and geodesic deviation depend on the masses of the DM halos and the typical length scales of the galaxies [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Due to the gravitational pull of DM halos, the apsidal precession of the geodesic orbits for EMRIs is strongly affected and even prograde-to-retrograde drift can occur [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' In prograde-to-retrograde orbital al- terations, GWs show transient frequency phenomena around a critial non-precessing turning point [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' A fully relativistic formalism to study GWs from EMRIs in static, spherically 3 symmetric spacetimes describing a MBH immersed in generic astrophysical environments was established in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' [57] and it was shown how the astrophysical environment changes GW generation and propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The above discussions are based on circular motions or eccentric cases without GW reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' In this paper, we study eccentric orbital motions and GWs of EMRIs in galaxies with DM environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' A review of the spacetime of galaxies harboring MBHs is given first, then we discuss the geodesic motions of EMRIs in the spacetime in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' In Section III, we use the ”Numerical Klugde” method [58– 60] to calculate GWs from eccentric EMRIs in galaxies with DM environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' To assess the capability of detecting DM halos with LISA, we calculate the mismatch between GWs from EMRIs with and without DM halos along with their signal-to noise (SNR) ratios in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' We draw conclusions in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' In this paper we use the units G = c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' THE MOTIONS OF BINARIES IN THE ENVIRONMENTS OF GALAXIES Following [50], we use the Hernquist-type density distribution [34] to describe the profiles observed in the bulges and elliptical galaxies ρH = Mr0 2πr(r + r0)3, (1) where M is the total mass of the DM halo, and r0 is the typical lengthscale of a galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The energy-momentum tensor of a galaxy harboring a MBH with the mass MBH is assumed to be an anisotropic fluid T µ ν = diag(−ρDM, 0, Pt, Pt), (2) where the density profile for a MBH residing at the center of the distribution (1) is 4πρDM = m′ r2 = 2M(r0 + 2MBH)(1 − 2MBH/r) r(r + r0)3 , (3) the mass function m(r) is m(r) = MBH + Mr2 (r0 + r)2 � 1 − 2MBH r �2 , (4) and the tangential pressure Pt is 2Pt = m(r)ρDM r − 2m(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' (5) 4 Obviously, in the absence of the MBH, the density profile (3) reduces to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' At large distance, r ≫ MBH, the density profile ρDM becomes the Hernquist-type distribution (1) for large galaxies with r0 ≫ MBH, ρDM ∼ (M/r0)2/(Mr), so the DM density ρDM is smaller if the compactness M/r0 is smaller with fixed M or if M is larger with fixed compactness M/r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Using the following ansatz for the static, spherically symmetric spacetime [50], ds2 = −f(r)dt2 + dr2 1 − 2m(r)/r + r2(dθ2 + sin2 θ dφ2), (6) and solving Einstein equations, we get [50] f(r) = � 1 − 2MBH r � eΥ, Υ = −π � M ξ + 2 � M ξ arctan �r + r0 − M √Mξ � , ξ = 2r0 − M + 4MBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' (7) The geometry (6) describes a BH spacetime with an horizon at r = 2MBH and a curvature singularity at r = 0, the matter density vanishes at the horizon and the ADM mass of the spacetime is M + MBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' In the absence of DM halo, M = 0, the spacetime (6) reduces to Schwarzschild BH with mass MBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' In galaxies, the compactness M/r0 can be as large as 10−4 [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' In general astrophysical environments the compactness M/r0 is usually small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Expanding the function f(r) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' (7) about M/r0 = 0 to the second order we get f(r) ≃ � 1 − 2MBH r � � 1 − 2M r0 + 4M 2 3r2 0 + 2Mr r2 0 + O[r−3 0 ] � = � 1 − 2MBH r � (1 + α + rβ), (8) where α = −2M/r0 + 4M 2/3r2 0 and β = 2M/r2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Now we consider a MBH in the center of a DM halo and a SCO moving on geodesics around the MBH in the equatorial plane (θ = π/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The geodesic equation is duµ dτ = 1 2uαuβ∂µgαβ, (9) where uα = drα/dτ, τ is the proper time and rα = (t, r, θ, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Because the spacetime is static and spherically symmetric, from the geodesic equation (9) we obtain two conserved quantities u0 = −E/µ and uφ = L/µ, u0 = −E/µ = − √ 1 + 2ε, (10) uφ = L/µ = h, (11) 5 where E and L represent the orbital energy and angular momentum of the system, respec- tively, and the reduced mass µ is approximately equal to the mass of the SCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The radial equation of motion is 1 + �dr dτ �2 � 1 − 2m(r) r �−1 + h2 r2 = 1 + 2ε f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' (12) For convenience, we introduce the orbital elements, the semi-latus rectum p and the eccentricity e, to parameterize the orbital motion, r = p 1 + e cos χ, (13) where χ is a parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Rewriting the variables h and ε in terms of p and e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' we obtain h2 = p Rs (1 + α) + p3β (1 − e2)−1 2(1 + α) � 1 − 1 2 Rs p (3 + e2) � + p β � 1 − 2 Rs p �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' (14) ε = − Rs 2p (1 − e2) � 1 − 2Rs p � + α j + α2 g + β k 2 � 1 − 1 2 Rs P (3 + e2) � (1 + α) + p β � 1 − 2 Rs p �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' (15) where Rs = 2MBH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' j = − � 1 − 2Rs p � + Rs 2p � 1 − 4Rs p � (1 − e2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' g = − � 1 − 2Rs p � − R2 s p2 (1 − e2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' k = −p(3 + e2) 2(1 − e2) � 1 − 2Rs p � − 2R2 s p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' In terms of χ, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' (10) and (11) become dφ dχ = �1 2 Rs p (1 + α) + 1 2pβ(1 − e2)−1 � 1 2 �1 2 Rs p � 1 − Rs p (3 + e cos χ) � + α A + 2α2 A + β B �− 1 2 J1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' (16) dt dχ = p (1 + e cos χ)2 �� 1 − (1 + e)Rs p � � 1 − (1 − e)Rs p � + C � 1 2 × � 1 − Rs p (1 + e cos χ) �−1 �1 2 Rs p � 1 − Rs p (3 + e cos χ) + αA + 2α2A + βB ��− 1 2 J2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' (17) 6 where A = Rs p � 1 − Rs p (3 + e cos χ) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' B = p 2(1 − e2)(1 + e cos χ) � 2 � 1 − Rs p � + � 1 − 4Rs p − �Rs p �2 (1 − e2)(1 + e cos χ) − Rs p e2(1 + cos2 χ) �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' C = α � 1 − 1 2(3 + e2)Rs p � + 1 2pβ � 1 − 2Rs r � − (αj + α2g + βk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' J1 = � 1 + α + βp 1 + e cos χ � 1 2 � 1 − 2Mp/(1 + e cos χ) a + p/(1 + e cos χ)2 � 1 − Rs p (1 + e cos χ) � �− 1 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' J2 = � 1 + α + βp 1 + e cos χ �− 1 2 � 1 − 2Mp/(1 + e cos χ) a + p/(1 + e cos χ)2 � 1 − Rs p (1 + e cos χ) � �− 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' (16) and (17) can be integrated to obtain φ(χ) and t(χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Taking different compact- ness and mass for the DM halo, using Cartesian coordinate (x, y) = (r cos φ, r sin φ) in the equatorial plane, we show the orbits of EMRIs in galaxies with and without DM in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Due to the gravitational drag of DM halos, the orbits with DM halos are different from those without DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' 1, we see that for the same value of M, the effect of DM halos on the orbital precession is larger if the compactness of the DM halo M/r0 is bigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' DM halos decrease the orbital precessions, and can even reverse the direction of precession if the density of DM halo ρDM is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The result of retrograde precessions of the orbital motion in the spacetime (6) is consistent with that found in [56], and the anomalous precessions of binaries in DM environments were also found in [48, 61, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' To probe DM halos and study their impact on the orbits of EMRIs, we calculate the time P and the orbital precession ∆φ over one cycle when the orbital parameter χ increases by 2π, T = � 2π 0 dt dχdχ, (18) ∆φ = � 2π 0 dφ dχdχ − 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' (19) Expanding Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' (16) and (17) about Rs/p = 0 to the second order and substituting the 7 100 50 0 50 100 100 50 0 50 100 x/Rs y/Rs r0=102M, M=102MBH 100 50 0 50 100 50 0 50 x/Rs y/Rs r0=103M, M=102MBH 100 50 0 50 100 50 0 50 x/Rs y/Rs r0=102M, M=103MBH 100 50 0 50 100 100 50 0 50 100 x/Rs y/Rs r0=103M, M=103MBH FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The orbits of EMRIs in galaxies with and without DM halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The mass of MBHs is set as MBH = 106M⊙, the eccentricity e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='6, and the semi-latus rectum p = 20Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' We take the compactness M/r0 as 10−2 and 10−3, and the total mass M as 102MBH and 103MBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The red dashed lines show the trajectories with DM and the blue solid lines show the orbits without DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The arrows represent the directions of orbital precessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' 8 results into Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' (18) and (19), we get T = 2π � 2p3 Rs 1 (1 − e2)3/2 � 1 + 3 2(1 − e2)Rs p + 3 2(1 − e2) � 1 + 5 4(1 − e2) 1 2 � �Rs p �2 + M r0 + 5M 2 6r2 0 + Mp r2 0(1 − e2) � e2 − 11 2 � − 3Mp2/Rs r2 0(1 − e2) � , (20) ∆φ = 3πRs p + 3π 8 (18 + e2) �Rs p �2 − 2π 1 − e2 Mp r2 0 � 3 + 1 + e2 + 2 Rs p (1 − e2)1/2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' (21) The terms with M in the above Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' (20) and (21) come from DM halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' In the absence of DM, M = 0, the above results (20) and (21) recover those for EMRIs with the central MBH being a Schwarzschild BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The dominant contribution to the period T in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' (20) is the first term, so T becomes larger as the semi-latus rectum p increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' However, there are positive and negative contributions from the local DM halos, the local DM halos may slow down the increase of T as p increases because the negative contribution in the last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' (20) and the presence of DM halos helps the increase of T with p if the last negative contribution is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' (21), it is easy to understand that the presence of DM halo decreases the orbital procession and even retrogrades the orbital procession if the local density of DM halos ρDM ∼ M/r2 0 is large enough so that the third term dominates over the first two terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' As the orbit becomes larger, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=', the semi-latus rectum p increases, the orbital precession decreases and the prograde precession decreases faster in the presence of DM halos because the third term due to DM halos in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' (21) becomes bigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' With DM halos, the prograde- to-retrograde precession transition happens at some critial value of p and then the prograde precessions change to retrograde precessions as p increases further;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' afterwards, the retrograde precessions increase as p increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Choosing different values for the compactness M/r0 and the total mass of DM halos M and using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' (20) and (21), we plot the results of the period T and the orbital precession ∆φ versus the semi-latus rectum p in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' As expected, the orbital period T increases with p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' the prograde precessions decrease with p and DM halos help the decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' For the case of r0 = 102M and M = 102MBH, the periapsis shifts change from prograde precessions to retrograde precessions at p = 60Rs and the retrograde precession increases with p when p ≳ 60Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' From the above discussions, we see that the orbital motions of EMRIs are influenced by DM halos, and we expect that the effects of local DM halos will leave imprints on GWs so that we can probe local DM halos through the observations of GWs emitted from EMRIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' 9 20 40 60 80 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='0 p/RS <Δϕ> 90 91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='11 r0=103M, M=103MBH r0=102M, M=103MBH r0=103M, M=102MBH r0=102M, M=102MBH M=0 30 40 50 60 70 80 90 100 10 20 30 40 50 p/Rs P/hour 90 91 41 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The results of orbital period and precession for EMRIs in galaxies with and without DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The mass of central MBHs is set as MBH = 106M⊙ and the eccentricity e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' We take the compactness M/r0 as 10−2 and 10−3, and the total mass M as 102MBH, 103MBH and M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The inserts show the evolution in a short time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' GWS OF EMRIS IN THE ENVIRONMENTS OF GALAXIES Using the above results for the orbital motions of EMRIs, we get the leading order energy and angular momentum fluxes �dE dt � GW ≃ 32 5 � µ MBH �2 �MBH p �5 (1 − e2)3/2 � 1 + 73 24e2 + 37 96e4 � � 1 − 6M r0 � , (22) �dL dt � GW ≃ 32 5 � µ MBH �2 MBH �MBH p �7/2 (1 − e2)3/2 � 1 + 7 8e2 � � 1 − 5M r0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' (23) The last factors 1 − 6M/r0 and 1 − 5M/r0 are the corrections from DM halos around the MBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Note that the effects of environmental DM halos on the losses of energy and angular momentum only depend on the compactness M/r0 and the energy and angular momentum fluxes become smaller if the compactness is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' In the absence of local DM halos, M = 0, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' (22) and (23) recover the standard results for eccentric binaries [63, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Applying the energy and angular momentum balance equations �dE dt � GW = − �dE dt � orbit , (24) �dL dt � GW = − �dL dt � orbit , (25) 10 we get the leading order evolution of the orbital parameters p(t) and e(t) due to the emission of GWs, dp dt = −64 5 µ MBH �MBH p �3 � 1 − e2� 3 2 � 1 + 7 8e2 � � 1 − 5M r0 � , (26) de dt = −304 15 e p µ MBH �MBH p �3 � 1 − e2� 3 2 � 1 + 121 304e2 � � 1 − 5M r0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' (27) Since the right sides of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' (26) and (27) are negative, both the semi-latus rectum p and the eccentricity decrease with time due to the radiation of GWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The presence of local DM halos slows down the decrease of p and e, the bigger the compactness M/r0 is, the slower the semi-latus rectum p(t) and the eccentricity decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' 3, we show the evolution of the orbital parameters p(t) and e(t) due to the emission of GWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Comparing with the astrophysical environments without DM, it takes more time for EMRIs with DM halos to evolve from p = 20Rs to p = 3Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The larger the compactness M/r0 is, the more time it takes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The presence of DM halos also slows down the decrease rate of the eccentricity and the final eccentricity is a bit larger with larger compactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' r0=102M, e0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='6 r0=103M, e0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='6 r0=102M, e0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='2 r0=103M, e0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='2 M=0, e0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='6 M=0, e0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='2 0 200 400 600 800 1000 0 5 10 15 20 t/yr p/Rs 0 200 400 600 800 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='6 t/yr e FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The evolution of the orbital parameters p and e from the initial p = 20Rs to p = (3+e)Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The mass of central MBHs is chosen as MBH = 106M⊙, the mass of the SCO is µ = 10M⊙ and the initial eccentricity is chosen as e0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' We consider two different values for the compactness of the DM halo, M/r0 = 10−2 and 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The solid lines correspond to the cases without DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' As discussed above, the effects of DM halos will be manifested in GW waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The quadrupole formula of GWs is hjk = 2 dL ¨Ijk, (28) 11 where dL is the luminosity distance between the detector and the source and Ijk is the quadrupole moment of EMRIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The tenser modes h+ and h× in the transverse-traceless gauge are given by h+ = 1 2 � ej Xek X − ej Y ek Y � hjk, (29) h× = 1 2 � ej Xek Y − ej Y ek X � hjk, (30) where eX and eY are the orthonormal vectors in the plane that is perpendicular to the direction from the detector to the GW source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Plugging the results for the orbital evolution obtained above into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' (28), we numerically calculate the time-domain GW waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The time-domain plus-mode GW waveforms for EMRIs with and without DM halos are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' From Fig 4, we see that initially the difference between GW waveforms with and without DM halos is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' One year later, the two waveforms for EMRIs with and without DM halos are quite different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' In order to quantify the impact of DM halo environments on the dephasing of GW waveforms, we calculate the number of orbital cycles accumulated from time ti to tf [65–67] N(t) = � tf ti ˙φ(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' (31) Over one-year evolution before the merger, the numbers of orbital cycles for EMRIs with and without DM halos are NDM and N0 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' In Fig 5, we show the difference ∆N = NDM − N0 between the number of orbital cycles with and without DM halos accumulated over one year before the merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Following [68], we choose ∆N ∼ 1 rad as the threshold for a detectable dephasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The results show that we can detect the compactness as small as ≲ 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The results also show that eccentric orbits can help detect DM halos with smaller compactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' To distinguish the waveforms more accurately, we calculate the mismatch between GW signals emitted from EMRIs with and without DM halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Given two signals h1(t) and h2(t), the inner product (h1|h2) is defined as (h1|h2) = 2 � +∞ 0 ˜h1(f)˜h∗ 2(f) + ˜h2(f)˜h∗ 1(f) Sh(f) df, (32) where ˜h(f) is the Fourier transformation of the time-domain signal h(t), ˜h∗ denotes the complex conjugate of ˜h, and the SNR for the signal h is � (h|h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' For LISA, the one-side 12 0 20 40 60 80 100 5 0 5 t/hour h+ At the beginning 0 20 40 60 80 100 5 0 5 t/hour h+ 365 days later r0=102M, M=102MBH M=0 1023× 1023× 0 20 40 60 80 100 5 0 5 t/hour h+ 0 20 40 60 80 100 5 0 5 t/hour h+ r0=103M, M=102MBH M=0 1023× 1023× FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The time-domain plus mode GW waveforms for EMRIs with and without DM halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The mass of central MBHs is MBH = 106M⊙, the mass of the SCO is µ = 10M⊙, the total mass of DM halos M is = 102MBH, the inclination angle ι = π/6, the luminosity distance dL = 1Gpc, the initial longitude of pericenter ω0 = 0 and the initial eccentricity e0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='6 at p0 = 20Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' M = 0 corresponds to the case without DM halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The left panels show the initial waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The right panels show the waveforms after one year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The top panels are for M/r0 = 10−2 and the bottom panels are for M/r0 = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' noise power spectral density is [69] Sh(f) = Sx L2 + 2Sa [1 + cos2(2π fL/c)] (2 πf)4L2 � 1 + �4 × 10−4Hz f �� , (33) where √Sa = 3 × 10−15 m s−2/Hz1/2 is the acceleration noise, √Sx = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='5 × 10−11 m/Hz1/2 is the displacement noise and L = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='5 × 106 km is the arm length of LISA [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The overlap between two GW signals is quantified as [60] O(˜h1, ˜h2) = (˜h1|˜h2) � (˜h1|˜h1)(˜h2|˜h2) , (34) ro=102M, M=102MBH M=0ro=103M, M=102MBH M-013 e0=0 e0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='2 e0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='4 e0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='6 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='5 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='5 2 300 250 200 150 100 50 0 Log10[M/r0] |Δ\uf77d| FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The difference between the orbital cycles with and without DM halos ∆N(t) over one-year evolution before the merger for different compactness of halos M/r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The initial eccentricity e0 is chosen at p0 = 20Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The mass of central MBHs is MBH = 106M⊙ and the mass of the SCO is µ = 10M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The masses of DM halos are M = 102MBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The black dashed line corresponds to ∆N = 1 rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' and the mismatch between two signals is defined as Mismatch = 1 − Omax(˜h1, ˜h2), (35) where the maximum is evaluated with respect to time and phase shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The mismatch is zero if two signals are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Two signals are considered experimentally distinguishable if their mismatch is larger than d/(2 SNR2), where d = 13 is the number of intrinsic parameters of the GW source [70–72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Considering EMRIs with masses (106+10)M⊙ at dL = 1 Gpc and integration time of one year before the coalescence, we calculate the mismatch between GW waveforms with and without DM halos and the results with LISA are shown in Fig 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The SNR is about 32 for the GW signals from EMRIs considered above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The initial eccentricity e0 is chosen at p0 = 20Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' As shown in Fig 6, if the compactness of DM halo M/r0 is larger, then the mismatch between GW waveforms with and without DM halos is bigger, so more compact DM halos can be detected easier with LISA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Again eccentric orbits can detect smaller compactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Therefore, we can use GWs from EMRIs in the environments of galaxies to test the existence of DM halos and detect the compactness of the halos M/r0 as small as 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' 14 e0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='2 e0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='6 6 5 4 3 2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='100 1 Log10[M/r0] Mismatch FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The results of the mismatch between GW waveforms with and without DM halos for different compactness M/r0 and initial eccentricity e0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The black dashed line corresponds to the threshold d/(2 SNR2) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='0072.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' CONCLUSIONS AND DISCUSSIONS Using the analytic, static and spherically symmetric metric for a Schwarzschild black hole immersed in DM halos with Hernquist type density distribution, we derive analytic formulae for the orbital period and orbital precession for eccentric EMRIs with the environment of DM halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The results show that the presence of DM halo decreases the orbital procession and even retrogrades the orbital procession if the local density of DM halos ρDM ∼ M/r2 0 is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' As the orbit becomes larger, the orbital precession decreases and the prograde precession decreases faster in the presence of DM halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' With DM halos, the prograde-to- retrograde precession transition happens at some critial value of p and then the prograde precessions change to retrograde precessions as p increases further;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' afterwards, the retrograde precessions increase as p increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Taking the energy and angular momentum fluxes of GWs into consideration, we derive analytic formulae for the evolutions of the semi-latus rectum and the eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' The presence of local DM halos slows down the decrease of the semi-latus rectum and the eccen- tricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Comparing the numbers of orbital cycles with and without DM halos over one-year evolution before the merger, we find that DM halos with the compactness as small as 10−4 can be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' By calculating the mismatch between GW waveforms with and without DM halos, we show that we can use GWs from EMRIs in the environments of galaxies to 15 test the existence of DM halos and detect the compactness as small as 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' We also find that eccentric orbits can help detect DM halos with smaller compactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Binaries in the environments of galaxies are also affected by the dynamical frictions of the surrounding medium [73–77], and the accretion of the medium [46, 78, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' It is necessary to consider the effects of dynamical frictions and accretion when the medium is dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' To distinguish the effects of DM halos from other mediums (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' accretion disks), or modified gravity on GWs, further study is needed [43, 68, 80–82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' ACKNOWLEDGMENTS The computing work in this paper is supported by the Public Service Platform of High Performance Computing by Network and Computing Center of HUST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' This research is supported in part by the National Key Research and Development Program of China under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' 2020YFC2201504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' (LIGO Scientific, Virgo), Observation of Gravitational Waves from a Binary Black Hole Merger, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' 116, 061102 (2016), arXiv:1602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='03837 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' [2] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' (LIGO Scientific, Virgo), GW150914: The Advanced LIGO Detectors in the Era of First Discoveries, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' 116, 131103 (2016), arXiv:1602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content='03838 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfdAy1/content/2301.05088v1.pdf'} +page_content=' [3] B.' metadata={'source': 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a/8tAyT4oBgHgl3EQfc_f6/content/tmp_files/2301.00296v1.pdf.txt b/8tAyT4oBgHgl3EQfc_f6/content/tmp_files/2301.00296v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bad687a780104720f173872d41413aab3822fb12 --- /dev/null +++ b/8tAyT4oBgHgl3EQfc_f6/content/tmp_files/2301.00296v1.pdf.txt @@ -0,0 +1,579 @@ +Local Einstein relation for fractals +L. Padilla, J. L. Iguain +Instituto de Investigaciones F´ısicas de Mar del Plata (IFIMAR) and +Departamento de F´ısica FCEyN, Universidad Nacional de Mar del Plata, +De´an Funes 3350, 7600 Mar del Plata, Argentina +E-mail: iguain@mdp.edu.ar +Abstract. +We study single random walks and the electrical resistance for fractals +obtained as the limit of a sequence of periodic structures. In the long-scale regime, +power laws describe both the mean-square displacement of a random walk as a function +of time and the electrical resistance as a function of length. +We show that the +corresponding power-law exponents satisfy the Einstein relation. For shorter scales, +where these exponents depend on length, we find how the Einstein relation can be +generalized to hold locally. All these findings were analytically derived and confirmed +by numerical simulations. +Keywords: Fractals, Power-law behaviours, Einstein relation. +arXiv:2301.00296v1 [cond-mat.stat-mech] 31 Dec 2022 + +Local Einstein relation for fractals +2 +1. Introduction +Fractals are characterized by quantities that exhibit power-law behaviour in space or +time. More precisely, as scale invariance occurs for integer powers of a characteristic +length, pure power laws are modulated by logarithmic periodic functions, that describe +the departures from the main trend at intermediate scales. These modulations have +been the object of recent interest and considerable effort has been devoted toward +understanding the relation between log-periodicity and discrete-scale invariance [1–13]. +For a given fractal and some related observables, which show (modulated) power- +law behaviours, a problem of interest is to determine whether or not the exponents +associated with these quantities are independent. Sometimes we can expect a relation +as a consequence of underlying physical laws. +This is, for example, the case of the +mass m, the electric resistance R and the mean-square-displacement (MSD) ∆r2 for +a single random walker. On a fractal, the first two grow with length l as m(l) ∼ ldf +and R(l) ∼ lζ, while the last one grows with time t as ∆r2(t) ∼ t2/dw. The exponents +df, ζ and dw are known as the fractal, resistance and walk exponents, respectively, and +these power-law behaviours hold for scales large enough to ensure self-similarity. In an +d-dimensional euclidean space, the diffusion coefficient D and conductivity σ are related +by the Einstein equation [14] +σ = e2ρ +kBT D. +(1) +Here, D = limt→∞ ∆r2(t)/2t, ρ and e are the density and charge of mobile particles, T +is the temperature and kB is the Boltzmann constant. Equation (1) is one of the forms +of the fluctuation-dissipation theorem, and can be used together with simple scaling +heuristic arguments, to argue that the fractal, walk, and resistance exponents satisfy +the Einstein relation [14] +df = dw − ζ, +(2) +This property has been shown to hold asymptotically for some finitely ramified +fractals [15, 16]; which has been used to analyze the periodicity of the oscillations in +dynamic observables, in the first attempts to understand log-periodic modulation [17]. +Einstein relation was also investigated for random walks on weighted graphs [18], and, +more recently, for karst networks structures [19]. +A deterministic fractal can be obtained as the limit of a sequence of periodic +structures. In this procedure, the period increases at every step as Ln (n = 0, 1, 2, ...), +where L is a basic characteristic length scale. Self-similarity is manifested in power-law +behaviours, which occur for long enough scales. However, this does not always hold +for shorter lengths. Thus, the local slopes of the observables as a function of time or +length, in log-log scales, are variable quantities, which approach constant values only +asymptotically. +In this work we argue that the local fractal, walk, and resistance exponents are +related through an equation that generalizes (2). +This generalization is obtained + +Local Einstein relation for fractals +3 +analytically, following the steady-state method for the calculation of the effective +diffusion coefficients for periodic substrates [20]. To further strengthen our findings we +perform numerical simulations for two models of fractals; which confirm the theoretical +predictions. +The paper is organized as follows. In Sec. 2 we relate the diffusion coefficient and the +unit cell resistance for a periodic structure. In Sec. 3 we derive the Einstein relation for +self-similar systems. In Sec. 4 we generalize this relation for scale-dependent exponents. +In Sec. 5 we confirm the generalized relation by numerical simulations performed on +models of asymptotic self-similar substrates. Finally, we give our conclusions in Sec. 6. +2. Periodic systems +In this section we address the problem of the diffusion coefficient for a periodic substrate. +We follows the steady-state method developed in reference [20]. We start by introducing +the periodic substrate with unit cell of linear dimension l, schematized in figure 1, where +the points represent sites, and the arrows represent hopping rates. On this structure, +a mobile particle can jump between connected sites according to the hopping rates k′s +(for the sake of clarity only a few sites and arrows were highlighted). We focus on a +steady-state of non-interacting particles flowing with a constant current density j. +k +n +n +(f+1) +(f+1) +n +(f+1) +n +ns +t +v +u +k +n +n +(f) +(f) +(f) +n +(f) +n +n(f) +k +s +t +v +r +u +(f+1) +(f+1) +l +krs +st +k u +t +uv +k +k +k +r +rs +st +tu +uv +Figure 1. Two nearest-neighbor cells f and f +1, for a periodic substrate with linear +size period l. The points represent sites, which can be occupied by mobile particles. +The arrows represent hopping rates between pairs of sites. For clarity, only a few sites +and hopping rates were highlighted. n(f) +r +corresponds to the number of particles in the +internal site r of cell f +As shown in [20], this steady-state consists of a set of microscopic currents +distributed with the same periodicity as the substrate. In figure 1 two nearest-neighbor +(NN) unit cells are depicted schematically where, for example, n(f) +s +represents the +number of particles in site r (internal index) of cell f. +Because of the mentioned + +Local Einstein relation for fractals +4 +periodicity, we get that for given pair of connected sites with internal indices s and +t, +i(f) +rs = i(f+1) +rs +, +(3) +where i(f) +rs is the current from site s to site r in cell f. In addition, as hopping rates do +not depend on the cell either but only on the internal indices, the last equation can be +rewritten as +ksr(n(f) +s +− n(f) +r ) = ksr(n(f+1) +s +− n(f+1) +r +), +(4) +or +n(f+1) +s +− n(f) +s += n(f+1) +r +− n(f) +r . +(5) +Therefore, in the steady-state, the difference in the occupation number for a given +site and the equivalent site in a NN cell is the same for all sites. +The relation of the steady-state problem with the diffusion coefficient D is provided +by Fick’s law +j = −D∆n +l2 , +(6) +which is valid for distances larger than l. Here ∆n corresponds to the particle number +difference for NN cells. Note that D also determines the mean-square displacement ∆2x +of a single random walker on the same structure, which behaves as +∆2x(t) = 2Dt; +(7) +for time long enough for ∆x ≫ l. +n +n +k +a +b +i=(n −n )k +b +a +Va +R +b +Vb +i= (V −V )/R +a +Figure 2. Schematics of the equivalence between Fick’s law (left) and Ohm’s law +(right). In the mapping particles have unitary charge, while the other quantities are +related as V = n, and R = 1/k. +Transforming the steady-state problem into an equivalent electrical problem is +straightforward. Indeed, for particles of unitary electric charge, a mapping between +Fick’s law and Ohm’s law results by identifying particle number with electrostatic +potential (Va = na) and hopping rate with conductance (k = 1/R). In figure 2 we +represent this mapping for every pair of connected sites. Following this analogy, we see + +Local Einstein relation for fractals +5 +that in the electric problem, the potential difference for a pair of equivalent sites in NN +cells takes the constant value +∆V = n(i+1) +r +− n(i) +r , +(8) +and that the difference between particle populations +∆n = +M +� +r=1 +(n(i+1) +r +− n(i) +r ) = M∆V, +(9) +is proportional to the potential difference ∆V , where the constant of proportionality M +corresponds to the number of sites per unit cell. +Thus, according to equation (6), we can conclude that, given a periodic substrate +with unit cell of linear dimension l and M sites, the diffusion coefficient and the potential +difference between two equivalent sites in NN cells, are connected through the relation +D = −j +l2 +M∆V , +(10) +where j is the steady-state current density. +3. Self-similar substrates +Deterministic fractals are usually built by a recursive procedure, that results in a +sequence of structures called generations. +A generation consists of a periodic array +of sites connected by bonds. The process begins with a basic periodic structure (zeroth +generation). At every step the unit cell is scaled by a factor L and the building rules +ensure that self-similarity is obtained after a large number of iterations. +Following equation (10), the diffusion coefficient Dp for the generation p and the +potential difference ∆Vp between two equivalent points in NN unit cells are related as +Dp = −j +L2p +Mp∆Vp +, +(11) +where Mp is the number of sites in the unit cell, and Lp is its linear dimension. Then, for +two consecutive generations p and p + 1, through which the same steady-state current +flows, we obtain +Dp +Dp+1 += L−2Mp+1 +Mp +∆Vp+1 +∆Vp +. +(12) +Now, since for a fractal the number of sites in a box with linear dimension l +behaves as m(l) ∼ ldf (i. e., df is the fractal dimension defined through box-counting), +Mp+1/Mp = (L(p+1)/Lp)df = Ldf, and the last equation can be rewritten as +Dp +Dp+1 += Ldf−2∆Vp+1 +∆Vp +, +(13) + +Local Einstein relation for fractals +6 +As previously shown [7,8], a perfect diffusive self-similar structure corresponds to +a ratio Dp/Dp+1 which does not depend on p, i. e., +Dp +Dp+1 += 1 + λ, +(14) +with λ a positive constant. In this model, the mean-square displacement for a single +random walker behaves as +∆2x(t) = f(t)t2ν. +(15) +The modulation f(t) is a log-periodic function, f(tτ) = f(t), and both ν and τ can be +analytically calculated in terms of L and λ: +ν = +1 +2 + log(1 + λ) +log(L) +(16) +τ = L1/ν +(17) +The important partial conclusion in the context of this work is that, according to +above discussion, a perfect diffusive self-similar structure implies a power-law behaviour +for the resistance as a function of length. Indeed, equations (13) and (14) leads to +∆Vp+1 +∆Vp += L1/ν−df, +(18) +where we have used 1 + λ = L1/ν−2, from equation (16). Thus, for a perfect diffusive +self-similar fractal the potential difference, which corresponds to steady-state current, +scales with length l as +∆V ∼ lζ, +(19) +where the exponent ζ is given by +ζ = 1/ν − df; +(20) +which is the Einstein relation (2), with dw = 1/ν. +4. Local exponents +We consider now a generic substrate for which diffusive self-similarity is reached only +asymptotically. Let us assume a ratio between consecutive diffusion coefficients, that +depends on the generation p, as +Dp +Dp+1 += 1 + λp. +(21) +where, {λp : p = 1, 2, ...} is a sequence of non-negative real numbers, with lim +p→∞ λp = λ. +Because of this limit, at long enough times a single random walk on this substrate +will show a MSD behaviour as in equation (15), and, as pointed out before, for large + +Local Einstein relation for fractals +7 +enough lengths the potential difference will behave as in equation (19); with ν and ζ +given by equations (16) and (20). +In this section we focus on local exponents, which correspond to the slopes in log- +log scales for finite length or time. As shown for example in [8], on a substrate on which +diffusion coefficients for generations p and p + 1 satisfy equation (21), the MSD for a +single random walker behaves as +∆2x(t) ∼ t2νp, +for +Lp ≲ ∆x ≲ Lp+1, +(22) +with the local exponent νp given by +νp = +1 +2 + log(1 + λp) +log(L) +· +(23) +Then, after rearranging this equation as 1+λp = L1/νp−2, which corresponds to the +left hand side of equation (13), we obtain +∆Vp+1 +∆Vp += L1/νp−df. +(24) +Thus, we expect that the potential difference scales with length l as +∆V (l) ∼ lζp, +for +Lp ≲ l ≲ Lp+1, +(25) +and that the local exponents satisfy the relation +ζp = 1/νp − df. +(26) +Therefore, local slopes in log-log scales for the resistance as a function of length +and for MSD of a single random walker as a function of time are related for all scales +through equation (26); which generalizes the Einstein relation. +5. Numerical simulations +We study numerically the steady-state that corresponds to a unitary current on two +models, for which diffusive self-similarity appears asymptotically. +At finite lengths, +the local random-walk exponent νp is not constant. Thus, we expect an also variable +resistance exponent ζp, related to the former through equation (26). +The first model is a substrate built on a square lattice. A random walk consists in +a particle hopping among NN sites. If sites are connected by a bond, the hopping rate is +k = 1/4. If the sites are not connected, the hopping rate is k = 0. A fractal is obtained +by deleting some bonds. The characteristic scale factor is L = 3, and the unit cells for +the first, the second and the third generations are depicted schematically in figure 3. +For every generation the unit cell can be separated from the rest by cutting four bonds. +As shown in a previous work, the mass on this structure shows a power-law behaviour +with df = 2. However, the random walk exponent νp grows with time and approaches +a value ν < 1/2 when t → ∞ [8]. + +Local Einstein relation for fractals +8 +We have run numerical simulations on the unit cell of the sixth generation, to reach +the steady-state in which a unitary current flows between the left and right extremes. In +figure 4 we plot with symbols the potential differences for lengths x = 3i (i = 0, 1, ..., 6), +which are the unit cell linear sizes for the generations zero to six. In the same figure, +we plot a line using the relation (26) and the numerical values for νp, which are the +outcomes of random walk simulations reported in reference [8]. Notice that both data +set fall on the same curve, which confirms the relation (26). +Figure 3. Substrate in two dimensions, which results in scale-dependent walk and +resistance exponents. The schematics correspond to the unit cells for the first, second +and third generations. The segments represent bonds between sites. +The second model is a generalization of the one-dimensional self-similar model +introduced in [7]. We start with a single random walk on a one-dimensional lattice, with +a hopping rate k0 between any pair of NN sites. This homogeneous case corresponds to +generation zero. We introduce a natural number L to build the other generations. +In the first generation, we reset to k1 < k0 the hopping rate for every pair of sites j +and j +1, with mod(j, L) = 0. The other hopping rates remains as in zeroth generation. +In the second generation, we reset to k2 < k1 the hopping rate for every pair of sites +j and j +1, with mod(j, L2) = 0. The other hopping rates remains as in first generation. +This recursion follows indefinitely, in such a way that generation n is obtained from +generation n − 1 after resetting to kn < kn−1 the hopping rate for every pair of sites j +and j + 1, with mod(j, Ln) = 0. In figure 5 we show an schematics for L = 5. + +Local Einstein relation for fractals +9 +1 +10 +100 +1 +10 +100 +1000 +∆V +x +Figure 4. Potential difference as a function of length for a unitary current flowing +trough the unit cell of the sixth generation substrate in figure 3. +The symbols +correspond to simulations of the steady-state. The line was plotted with the exponents +ζp from equation (26) and the values of νp which result from random-walk numerical +simulations. +L +L2 +k0 +k1 +k2 +Figure 5. Schematics of the one-dimensional random-walk model. We begin with +a homogeneous lattice, and a hopping rate k0 between nearest-neighbor sites. Then, +hopping rates are reset to kj for transitions between sites j and j + 1 for every j such +that mod(j, Ln) = 0, and for n = 1, 2, .... In this example, L = 5. +If we ask for perfect self-similarity for diffusion, i. e. equation (14), the hopping +rates are found iteratively as in reference [7]. For the more general case of equation +(21), the sequence of hopping rates is given by +1 +ki += +1 +ki−1 ++ Liλi−1 +k0 +i−2 +� +j=0 +(1 + λj), +for i = 1, 2, 3... +(27) +We test the validity of the relation (26) among the local exponents for a family of + +Local Einstein relation for fractals +10 +substrates given by +λp = λ (1 − 2−p/5.). +(28) +At short enough lengths these substrates are nearly homogeneous (λp ≈ 0 for p ≪ 5), +while, on the other extreme, self-similarity for diffusion is reached for lengths much larger +than L5. The local random walk exponent (23) decreases with length and approaches +asymptotically ν in equation (16). Thus, the variation of νp in space increases with λ +and, because of equation (26), the same should occur with the variation of ζp. This is an +interesting model, because the variation of the exponents with length can be adjusted +through the parameter λ. +100 +101 +102 +103 +104 +105 +106 +107 +108 +100 +101 +102 +103 +100 +101 +102 +103 +104 +105 +106 +100 +101 +102 +103 +∆V +x +∆V +x +Figure 6. Potential difference as a function of length for unitary current on the one- +dimensional model with λp = λ (1−2−p/5.), and L = 2. (Main) Symbols correspond to +data obtained with numerical simulations on a tenth-generation substrate. Lines were +drawn using the values of theoretical exponents. From bottom to top, λ = 1 (red), +λ = 2 (green), λ = 4 (violet), λ = 5 (blue). (Inset) More detailed structure for λ = 2. +We have run numerical simulations for the steady-state that corresponds to a +unitary current flowing on this model, with L = 2 and λ = 1, 2, 4, 5. All substrates +were built until generation 10. In figure 6-main we plot with symbols the potential +difference as a function of the length x, for x = 2j (j = 0, 1, ..., 9). The lines correspond +to the exponents ζp obtained from equations (26) and (23). Note the excellent agreement +between theory and simulations. The inset in the same figure shows substructure of ∆V +for λ = 2. +6. Conclusions +We have studied first the connection between single random walks and steady-state +potential difference for substrates with spatial periodicity. +Then, by considering a +sequence of periodic systems, a common procedure for deterministic fractal construction, +we find that the length dependent fractal, walk and resistance exponents, for the + +Local Einstein relation for fractals +11 +substrate obtained in the infinite limit of this sequence, satisfy, at every length scale, +the relation (26). This can be considered as a local version of the Einstein relation (2). +We have tested our predictions numerically for two models. The first model is a fractal +in two dimensions, while the the second is a fractal in one dimension. Both models lead +to length-dependent exponents at intermediate scales. The excellent agreement between +the outcomes of these simulations and the theoretical predictions supports the validity +of the mentioned relation among exponents, not only in the asymptotic self-similar limit +but also locally, for all length scales. +Acknowledgments +We are grateful to H. O. M´artin for useful discussions. This research was supported +by the Universidad Nacional de Mar del Plata, 15/E1040, and the Consejo Nacional de +Investigaciones Cient´ıficas y T´ecnicas, PIP1748/21. +References +[1] Peter J. Grabner and Wolfgang Woess. Functional iterations and periodic oscillations for simple +random walk on the sierpi?ski graph. Stochastic Processes and their Applications, 69(1):127 – +138, 1997. +[2] L. Acedo and S. B. Yuste. Territory covered by n random walkers on fractal media: The sierpinski +gasket and the percolation aggregate. Phys. Rev. E, 63:011105, Dec 2000. +[3] M. A. Bab, G. Fabricius, and E. V. Albano. On the occurrence of oscillatory modulations in the +power law behavior of dynamic and kinetic processes in fractals. EPL (Europhysics Letters), +81(1):10003, 2008. +[4] M. A. Bab, G. Fabricius, and Ezequiel V. Albano. Revisiting random walks in fractal media: On +the occurrence of time discrete scale invariance. +The Journal of Chemical Physics, 128(4):–, +2008. +[5] Alberto L. Maltz, Gabriel Fabricius, Marisa A. Bab, and Ezequiel V. Albano. +Random walks +in fractal media: +a theoretical evaluation of the periodicity of the oscillations in dynamic +observables. Journal of Physics A: Mathematical and Theoretical, 41(49):495004, 2008. +[6] Sebastian Weber, Joseph Klafter, and Alexander Blumen. Random walks on sierpinski gaskets of +different dimensions. Phys. Rev. E, 82:051129, Nov 2010. +[7] L. 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A., Padilla, L., M´artin, H. O., and Iguain, J. L. +Intermediate-range structure in +ion-conducting tellurite glasses. EPL, 103(3):36002, 2013. +[14] Amin Bunde and Shlomo Havlin (Eds.). Fractals and Disordered Systems. Springer, 1996. + +Local Einstein relation for fractals +12 +[15] J A Given and B B Mandelbrot. +Diffusion on fractal lattices and the fractal einstein relation. +Journal of Physics A: Mathematical and General, 16(15):L565, oct 1983. +[16] Astrid Franz, Christian Schulzky, and Karl Heinz Hoffmann. +The Einstein relation for finitely +ramified Sierpinski carpets. Nonlinearity, 14(5):1411, aug 2001. +[17] Alberto L Maltz, Gabriel Fabricius, Marisa A Bab, and Ezequiel V Albano. +Random walks +in fractal media: +a theoretical evaluation of the periodicity of the oscillations in dynamic +observables. Journal of Physics A: Mathematical and Theoretical, 41(49):495004, oct 2008. +[18] Andr´as Telcs. The Einstein Relation for Random Walks on Graphs. Journal of Statistical Physics, +122(4):617–645, 2006. +[19] Martin Hendrick and Philippe Renard. +Fractal dimension, walk dimension and conductivity +exponent of karst networks around tulum. Frontiers in Physics, 4, 2016. +[20] C. M. Aldao, J. L. Iguain, and H. O. M´artin. Diffusion of tagged particle in an exclusion process. +Surf. Sci., 366:483–490, Apr 1996. + diff --git a/8tAyT4oBgHgl3EQfc_f6/content/tmp_files/load_file.txt b/8tAyT4oBgHgl3EQfc_f6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8309ef1d6fd9097708ce4bb6ced15edd27602376 --- /dev/null +++ b/8tAyT4oBgHgl3EQfc_f6/content/tmp_files/load_file.txt @@ -0,0 +1,310 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf,len=309 +page_content='Local Einstein relation for fractals L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Padilla, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Iguain Instituto de Investigaciones F´ısicas de Mar del Plata (IFIMAR) and Departamento de F´ısica FCEyN, Universidad Nacional de Mar del Plata, De´an Funes 3350, 7600 Mar del Plata, Argentina E-mail: iguain@mdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content='ar Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' We study single random walks and the electrical resistance for fractals obtained as the limit of a sequence of periodic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' In the long-scale regime, power laws describe both the mean-square displacement of a random walk as a function of time and the electrical resistance as a function of length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' We show that the corresponding power-law exponents satisfy the Einstein relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' For shorter scales, where these exponents depend on length, we find how the Einstein relation can be generalized to hold locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' All these findings were analytically derived and confirmed by numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Keywords: Fractals, Power-law behaviours, Einstein relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content='00296v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content='stat-mech] 31 Dec 2022 Local Einstein relation for fractals 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Introduction Fractals are characterized by quantities that exhibit power-law behaviour in space or time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' More precisely, as scale invariance occurs for integer powers of a characteristic length, pure power laws are modulated by logarithmic periodic functions, that describe the departures from the main trend at intermediate scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' These modulations have been the object of recent interest and considerable effort has been devoted toward understanding the relation between log-periodicity and discrete-scale invariance [1–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' For a given fractal and some related observables, which show (modulated) power- law behaviours, a problem of interest is to determine whether or not the exponents associated with these quantities are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Sometimes we can expect a relation as a consequence of underlying physical laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' This is, for example, the case of the mass m, the electric resistance R and the mean-square-displacement (MSD) ∆r2 for a single random walker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' On a fractal, the first two grow with length l as m(l) ∼ ldf and R(l) ∼ lζ, while the last one grows with time t as ∆r2(t) ∼ t2/dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' The exponents df, ζ and dw are known as the fractal, resistance and walk exponents, respectively, and these power-law behaviours hold for scales large enough to ensure self-similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' In an d-dimensional euclidean space, the diffusion coefficient D and conductivity σ are related by the Einstein equation [14] σ = e2ρ kBT D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' (1) Here, D = limt→∞ ∆r2(t)/2t, ρ and e are the density and charge of mobile particles, T is the temperature and kB is the Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Equation (1) is one of the forms of the fluctuation-dissipation theorem, and can be used together with simple scaling heuristic arguments, to argue that the fractal, walk, and resistance exponents satisfy the Einstein relation [14] df = dw − ζ, (2) This property has been shown to hold asymptotically for some finitely ramified fractals [15, 16];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' which has been used to analyze the periodicity of the oscillations in dynamic observables, in the first attempts to understand log-periodic modulation [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Einstein relation was also investigated for random walks on weighted graphs [18], and, more recently, for karst networks structures [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' A deterministic fractal can be obtained as the limit of a sequence of periodic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' In this procedure, the period increases at every step as Ln (n = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content='), where L is a basic characteristic length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Self-similarity is manifested in power-law behaviours, which occur for long enough scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' However, this does not always hold for shorter lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Thus, the local slopes of the observables as a function of time or length, in log-log scales, are variable quantities, which approach constant values only asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' In this work we argue that the local fractal, walk, and resistance exponents are related through an equation that generalizes (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' This generalization is obtained Local Einstein relation for fractals 3 analytically, following the steady-state method for the calculation of the effective diffusion coefficients for periodic substrates [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' To further strengthen our findings we perform numerical simulations for two models of fractals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' which confirm the theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' 2 we relate the diffusion coefficient and the unit cell resistance for a periodic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' 3 we derive the Einstein relation for self-similar systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' 4 we generalize this relation for scale-dependent exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' 5 we confirm the generalized relation by numerical simulations performed on models of asymptotic self-similar substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Finally, we give our conclusions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Periodic systems In this section we address the problem of the diffusion coefficient for a periodic substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' We follows the steady-state method developed in reference [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' We start by introducing the periodic substrate with unit cell of linear dimension l, schematized in figure 1, where the points represent sites, and the arrows represent hopping rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' On this structure, a mobile particle can jump between connected sites according to the hopping rates k′s (for the sake of clarity only a few sites and arrows were highlighted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' We focus on a steady-state of non-interacting particles flowing with a constant current density j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' k n n (f+1) (f+1) n (f+1) n ns t v u k n n (f) (f) (f) n (f) n n(f) k s t v r u (f+1) (f+1) l krs st k u t uv k k k r rs st tu uv Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Two nearest-neighbor cells f and f +1, for a periodic substrate with linear size period l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' The points represent sites, which can be occupied by mobile particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' The arrows represent hopping rates between pairs of sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' For clarity, only a few sites and hopping rates were highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' n(f) r corresponds to the number of particles in the internal site r of cell f As shown in [20], this steady-state consists of a set of microscopic currents distributed with the same periodicity as the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' In figure 1 two nearest-neighbor (NN) unit cells are depicted schematically where, for example, n(f) s represents the number of particles in site r (internal index) of cell f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Because of the mentioned Local Einstein relation for fractals 4 periodicity, we get that for given pair of connected sites with internal indices s and t, i(f) rs = i(f+1) rs , (3) where i(f) rs is the current from site s to site r in cell f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' In addition, as hopping rates do not depend on the cell either but only on the internal indices, the last equation can be rewritten as ksr(n(f) s − n(f) r ) = ksr(n(f+1) s − n(f+1) r ), (4) or n(f+1) s − n(f) s = n(f+1) r − n(f) r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' (5) Therefore, in the steady-state, the difference in the occupation number for a given site and the equivalent site in a NN cell is the same for all sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' The relation of the steady-state problem with the diffusion coefficient D is provided by Fick’s law j = −D∆n l2 , (6) which is valid for distances larger than l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Here ∆n corresponds to the particle number difference for NN cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Note that D also determines the mean-square displacement ∆2x of a single random walker on the same structure, which behaves as ∆2x(t) = 2Dt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' (7) for time long enough for ∆x ≫ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' n n k a b i=(n −n )k b a Va R b Vb i= (V −V )/R a Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Schematics of the equivalence between Fick’s law (left) and Ohm’s law (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' In the mapping particles have unitary charge, while the other quantities are related as V = n, and R = 1/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Transforming the steady-state problem into an equivalent electrical problem is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Indeed, for particles of unitary electric charge, a mapping between Fick’s law and Ohm’s law results by identifying particle number with electrostatic potential (Va = na) and hopping rate with conductance (k = 1/R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' In figure 2 we represent this mapping for every pair of connected sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Following this analogy, we see Local Einstein relation for fractals 5 that in the electric problem, the potential difference for a pair of equivalent sites in NN cells takes the constant value ∆V = n(i+1) r − n(i) r , (8) and that the difference between particle populations ∆n = M � r=1 (n(i+1) r − n(i) r ) = M∆V, (9) is proportional to the potential difference ∆V , where the constant of proportionality M corresponds to the number of sites per unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Thus, according to equation (6), we can conclude that, given a periodic substrate with unit cell of linear dimension l and M sites, the diffusion coefficient and the potential difference between two equivalent sites in NN cells, are connected through the relation D = −j l2 M∆V , (10) where j is the steady-state current density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Self-similar substrates Deterministic fractals are usually built by a recursive procedure, that results in a sequence of structures called generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' A generation consists of a periodic array of sites connected by bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' The process begins with a basic periodic structure (zeroth generation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' At every step the unit cell is scaled by a factor L and the building rules ensure that self-similarity is obtained after a large number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Following equation (10), the diffusion coefficient Dp for the generation p and the potential difference ∆Vp between two equivalent points in NN unit cells are related as Dp = −j L2p Mp∆Vp , (11) where Mp is the number of sites in the unit cell, and Lp is its linear dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Then, for two consecutive generations p and p + 1, through which the same steady-state current flows, we obtain Dp Dp+1 = L−2Mp+1 Mp ∆Vp+1 ∆Vp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' (12) Now, since for a fractal the number of sites in a box with linear dimension l behaves as m(l) ∼ ldf (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=', df is the fractal dimension defined through box-counting), Mp+1/Mp = (L(p+1)/Lp)df = Ldf, and the last equation can be rewritten as Dp Dp+1 = Ldf−2∆Vp+1 ∆Vp , (13) Local Einstein relation for fractals 6 As previously shown [7,8], a perfect diffusive self-similar structure corresponds to a ratio Dp/Dp+1 which does not depend on p, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=', Dp Dp+1 = 1 + λ, (14) with λ a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' In this model, the mean-square displacement for a single random walker behaves as ∆2x(t) = f(t)t2ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' (15) The modulation f(t) is a log-periodic function, f(tτ) = f(t), and both ν and τ can be analytically calculated in terms of L and λ: ν = 1 2 + log(1 + λ) log(L) (16) τ = L1/ν (17) The important partial conclusion in the context of this work is that, according to above discussion, a perfect diffusive self-similar structure implies a power-law behaviour for the resistance as a function of length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Indeed, equations (13) and (14) leads to ∆Vp+1 ∆Vp = L1/ν−df, (18) where we have used 1 + λ = L1/ν−2, from equation (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Thus, for a perfect diffusive self-similar fractal the potential difference, which corresponds to steady-state current, scales with length l as ∆V ∼ lζ, (19) where the exponent ζ is given by ζ = 1/ν − df;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' (20) which is the Einstein relation (2), with dw = 1/ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Local exponents We consider now a generic substrate for which diffusive self-similarity is reached only asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Let us assume a ratio between consecutive diffusion coefficients, that depends on the generation p, as Dp Dp+1 = 1 + λp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' (21) where, {λp : p = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content='} is a sequence of non-negative real numbers, with lim p→∞ λp = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Because of this limit, at long enough times a single random walk on this substrate will show a MSD behaviour as in equation (15), and, as pointed out before, for large Local Einstein relation for fractals 7 enough lengths the potential difference will behave as in equation (19);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' with ν and ζ given by equations (16) and (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' In this section we focus on local exponents, which correspond to the slopes in log- log scales for finite length or time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' As shown for example in [8], on a substrate on which diffusion coefficients for generations p and p + 1 satisfy equation (21), the MSD for a single random walker behaves as ∆2x(t) ∼ t2νp, for Lp ≲ ∆x ≲ Lp+1, (22) with the local exponent νp given by νp = 1 2 + log(1 + λp) log(L) (23) Then, after rearranging this equation as 1+λp = L1/νp−2, which corresponds to the left hand side of equation (13), we obtain ∆Vp+1 ∆Vp = L1/νp−df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' (24) Thus, we expect that the potential difference scales with length l as ∆V (l) ∼ lζp, for Lp ≲ l ≲ Lp+1, (25) and that the local exponents satisfy the relation ζp = 1/νp − df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' (26) Therefore, local slopes in log-log scales for the resistance as a function of length and for MSD of a single random walker as a function of time are related for all scales through equation (26);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' which generalizes the Einstein relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Numerical simulations We study numerically the steady-state that corresponds to a unitary current on two models, for which diffusive self-similarity appears asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' At finite lengths, the local random-walk exponent νp is not constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Thus, we expect an also variable resistance exponent ζp, related to the former through equation (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' The first model is a substrate built on a square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' A random walk consists in a particle hopping among NN sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' If sites are connected by a bond, the hopping rate is k = 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' If the sites are not connected, the hopping rate is k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' A fractal is obtained by deleting some bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' The characteristic scale factor is L = 3, and the unit cells for the first, the second and the third generations are depicted schematically in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' For every generation the unit cell can be separated from the rest by cutting four bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' As shown in a previous work, the mass on this structure shows a power-law behaviour with df = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' However, the random walk exponent νp grows with time and approaches a value ν < 1/2 when t → ∞ [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Local Einstein relation for fractals 8 We have run numerical simulations on the unit cell of the sixth generation, to reach the steady-state in which a unitary current flows between the left and right extremes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' In figure 4 we plot with symbols the potential differences for lengths x = 3i (i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=', 6), which are the unit cell linear sizes for the generations zero to six.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' In the same figure, we plot a line using the relation (26) and the numerical values for νp, which are the outcomes of random walk simulations reported in reference [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Notice that both data set fall on the same curve, which confirms the relation (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Substrate in two dimensions, which results in scale-dependent walk and resistance exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' The schematics correspond to the unit cells for the first, second and third generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' The segments represent bonds between sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' The second model is a generalization of the one-dimensional self-similar model introduced in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' We start with a single random walk on a one-dimensional lattice, with a hopping rate k0 between any pair of NN sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' This homogeneous case corresponds to generation zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' We introduce a natural number L to build the other generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' In the first generation, we reset to k1 < k0 the hopping rate for every pair of sites j and j +1, with mod(j, L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' The other hopping rates remains as in zeroth generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' In the second generation, we reset to k2 < k1 the hopping rate for every pair of sites j and j +1, with mod(j, L2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' The other hopping rates remains as in first generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' This recursion follows indefinitely, in such a way that generation n is obtained from generation n − 1 after resetting to kn < kn−1 the hopping rate for every pair of sites j and j + 1, with mod(j, Ln) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' In figure 5 we show an schematics for L = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Local Einstein relation for fractals 9 1 10 100 1 10 100 1000 ∆V x Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Potential difference as a function of length for a unitary current flowing trough the unit cell of the sixth generation substrate in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' The symbols correspond to simulations of the steady-state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' The line was plotted with the exponents ζp from equation (26) and the values of νp which result from random-walk numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' L L2 k0 k1 k2 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Schematics of the one-dimensional random-walk model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' We begin with a homogeneous lattice, and a hopping rate k0 between nearest-neighbor sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Then, hopping rates are reset to kj for transitions between sites j and j + 1 for every j such that mod(j, Ln) = 0, and for n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content='. In this example, L = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' If we ask for perfect self-similarity for diffusion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' equation (14), the hopping rates are found iteratively as in reference [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' For the more general case of equation (21), the sequence of hopping rates is given by 1 ki = 1 ki−1 + Liλi−1 k0 i−2 � j=0 (1 + λj), for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' (27) We test the validity of the relation (26) among the local exponents for a family of Local Einstein relation for fractals 10 substrates given by λp = λ (1 − 2−p/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' (28) At short enough lengths these substrates are nearly homogeneous (λp ≈ 0 for p ≪ 5), while, on the other extreme, self-similarity for diffusion is reached for lengths much larger than L5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' The local random walk exponent (23) decreases with length and approaches asymptotically ν in equation (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Thus, the variation of νp in space increases with λ and, because of equation (26), the same should occur with the variation of ζp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' This is an interesting model, because the variation of the exponents with length can be adjusted through the parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' 100 101 102 103 104 105 106 107 108 100 101 102 103 100 101 102 103 104 105 106 100 101 102 103 ∆V x ∆V x Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Potential difference as a function of length for unitary current on the one- dimensional model with λp = λ (1−2−p/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' ), and L = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' (Main) Symbols correspond to data obtained with numerical simulations on a tenth-generation substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Lines were drawn using the values of theoretical exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' From bottom to top, λ = 1 (red), λ = 2 (green), λ = 4 (violet), λ = 5 (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' (Inset) More detailed structure for λ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' We have run numerical simulations for the steady-state that corresponds to a unitary current flowing on this model, with L = 2 and λ = 1, 2, 4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' All substrates were built until generation 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' In figure 6-main we plot with symbols the potential difference as a function of the length x, for x = 2j (j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=', 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' The lines correspond to the exponents ζp obtained from equations (26) and (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Note the excellent agreement between theory and simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' The inset in the same figure shows substructure of ∆V for λ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Conclusions We have studied first the connection between single random walks and steady-state potential difference for substrates with spatial periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Then, by considering a sequence of periodic systems, a common procedure for deterministic fractal construction, we find that the length dependent fractal, walk and resistance exponents, for the Local Einstein relation for fractals 11 substrate obtained in the infinite limit of this sequence, satisfy, at every length scale, the relation (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' This can be considered as a local version of the Einstein relation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' We have tested our predictions numerically for two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' The first model is a fractal in two dimensions, while the the second is a fractal in one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Both models lead to length-dependent exponents at intermediate scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' The excellent agreement between the outcomes of these simulations and the theoretical predictions supports the validity of the mentioned relation among exponents, not only in the asymptotic self-similar limit but also locally, for all length scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Acknowledgments We are grateful to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' M´artin for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' This research was supported by the Universidad Nacional de Mar del Plata, 15/E1040, and the Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas, PIP1748/21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' References [1] Peter J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfc_f6/content/2301.00296v1.pdf'} +page_content=' Grabner and Wolfgang Woess.' metadata={'source': 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b/9NAzT4oBgHgl3EQfFPpp/content/tmp_files/2301.01007v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..94082ff61200147c970d6c10a2dbeeb50aff6c0b --- /dev/null +++ b/9NAzT4oBgHgl3EQfFPpp/content/tmp_files/2301.01007v1.pdf.txt @@ -0,0 +1,2688 @@ +A Bertrand duopoly game with differentiated products reconsidered +Xiaoliang Lia and Bo Li∗b +aSchool of Digital Economics, Dongguan City University, Dongguan 523419, China +bSchool of Finance, Anhui University of Finance and Economics, Bengbu 233030, China +Abstract +In this paper, we explore a dynamic Bertrand duopoly game with differentiated products, where +firms are boundedly rational and consumers are assumed to possess an underlying CES utility function. +We mainly focus on two distinct degrees of product substitutability. Several tools based on symbolic +computations such as the triangular decomposition method and the PCAD method are employed in the +analytical investigation of the model. The uniqueness of the non-vanishing equilibrium is proved and +rigorous conditions for the local stability of this equilibrium are established for the first time. +Most +importantly, we find that increasing the substitutability degree or decreasing the product differentiation +has an effect of destabilization for our Bertrand model, which is in contrast with the relative conclusions +for the Cournot models. This finding could be conducive to the revelation of the essential difference +between dynamic Cournot and Bertrand oligopolies with differentiated goods. +In the special case of +identical marginal costs, we derive that lower degrees of product differentiation mean lower prices, higher +supplies, lower profits, and lower social welfare. Furthermore, complex dynamics such as periodic orbits +and chaos are reported through our numerical simulations. +Keywords: Bertrand duopoly; differentiated product; symbolic computation; local stability +1 +Introduction +It is well known that Cournot [12] developed the first formal theory of oligopoly, which is a market supplied +by only a few firms. In Cournot’s framework, firms are supposed to make decisions on their quantities of +outputs and have perfect information on their rivals’ strategic behavior. In the strand of Cournot oligopoly +models, the market demand function is usually supposed to be linear for simplicity by many economists +(e.g., Fisher [16], McManus and Quandt[29]). In the real world, however, a non-linear demand is more +likely to exist. Puu [33] investigated a Cournot duopoly game under an isoelastic market demand, where +the price is simply the reciprocal of the total supply. Afterward, fruitful contributions including [2, 4, 7, +9, 10, 13, 20, 21, 22, 24, 28, 31], were made in the literature on Cournot games. Related to our study, +Zhang and Zhang [39] considered a Cournot game in which each firm produces multiple products and sells +them in multiple markets. They obtained sufficient and necessary conditions for the local stability of the +Cournot-Nash equilibria. +Several decades later after Cournot’s seminal work, Bertrand [6] proposed a different framework to +describe oligopolistic competition, where prices rather than quantities are the strategic variables of the +competitors. Singh and Vives [35] analyzed the duality of prices and quantities, and found that Cournot +(Bertrand) competition with substitutes is the dual of Bertrand (Cournot) competition with complements. +L´opez and Naylor [26] compared Cournot and Bertrand equilibria in a downstream differentiated duopoly, +and proved that the classic conclusion that profits under Cournot equilibrium exceed those under Bertrand +competition could be reversible in the case of imperfect substitutes. Zhang et al. [40] considered a Bertrand +model formulated under a linear inverse demand, and obtained the existence and stability of the equilibrium. +Different from [40], Fanti et al. [15] developed a model with sound microeconomic foundations that deter- +mine the demand for differentiated products, and showed that synchronized dynamics and intermittency +phenomena may appear. Naimzada and Tramontana [32] also considered a Cournot-Bertrand duopoly model +with product differentiation and emphasized the role of best response dynamics and an adaptive adjustment +mechanism for stability. Brianzoni et al. [8] assumed quadratic costs in the study of the Bertrand duopoly +∗Corresponding author: libomaths@163.com +1 +arXiv:2301.01007v1 [econ.TH] 3 Jan 2023 + +game with horizontal product differentiation and discovered synchronized dynamics. Moreover, Ma and Guo +[27] studied the impacts of information on the dynamical Bertrand game. They showed that there exists +a fixed point independent of the amount of information for a triopoly, and the stable region of adjustment +parameter increases with the amount of information for a duopoly. +In all the aforementioned Bertrand games, the inverse demand function is supposed to be linear. Instead, +Gori and Sodini [17] explored the local and global dynamics of a Bertrand duopoly with a nonlinear demand +and horizontal product differentiation. Furthermore, Ahmed et al. [3] proposed a dynamic Bertrand duopoly +game with differentiated products, where firms are boundedly rational and consumers are assumed to possess +an underlying CES utility function. They only employed numerical simulations to investigate the dynamic +behavior of their model because the closed form of the equilibrium is extremely difficult to compute. They +observed that the Nash equilibrium loses its stability through a period-doubling bifurcation as the speed +of adjustment increases. +Motivated by [3], Agliari et al. [1] investigated a Cournot duopoly game with +differentiated goods. We should mention that Agliari et al. [1] used the same CES utility function as [3] to +derive the demand function of the market. They discovered that a low degree of product substitutability or +a higher degree of product differentiation may destabilize the Cournot game. This finding is in accordance +with that of Fanti and Gori [14], where the authors introduced a Cournot duopoly with a linear demand and +heterogeneous players to study the influence of product differentiation on stability and found that a higher +degree of product differentiation may destabilize the market equilibrium. +In this paper, we re-study the Bertrand duopoly game of Ahmed et al. [3] using several tools based on +symbolic computations such as the triangular decomposition method (see, e.g., [23]) and the PCAD method +(see, e.g., [11]). It is worth noting that the results of symbolic computations are exact, and thus can provide +theoretical foundations for the systematic analysis of economic models. +We analytically investigate the +local stability and bifurcations of the model. By using several tools based on symbolic computations, the +uniqueness of the non-vanishing equilibrium is proved and the rigorous conditions for the local stability of this +equilibrium are obtained for the first time. In the special case that the two companies have identical marginal +costs, we prove that the model can lose its stability only through a period-doubling bifurcation. The most +important finding is that increasing the substitutability degree or decreasing the product differentiation has +an effect of destabilizing the unique non-vanishing equilibrium. A possible explanation is that a decrease in +product differentiation may result in an increase in market competition intensity and even a price war, which +could lead to the destabilization of the equilibrium. It should be noted that our finding is in contrast with +the relative conclusions by Agliari et al. [1] and by Fanti and Gori [14]. This contradiction contributes to +the literature on the connection between Cournot and Bertrand oligopolies and may help reveal the essential +difference between them. In the special case of identical marginal costs, we derive the fact that lower degrees +of product differentiation can lead to lower prices, higher supplies, lower profits, and lower social welfare. +This fact is in line with our economic intuition. Complex dynamics such as periodic orbits and chaos can +be observed through our numerical simulations, which also confirm that an increase in the substitutability +degree leads to the emergence of instability in the considered model. Furthermore, we discover the existence +of a Neimark-Sacker bifurcation directly on the equilibrium, which is a new finding and has not yet been +discovered by Ahmed et al. [3] +The rest of this paper is structured as follows. In Section 2, we revisit the construction of the Bertrand +duopoly game investigated in our study. We analytically explore the stability and bifurcations of this model +for two different substitutability degrees, namely α = 1/2 and α = 1/3, in Sections 3 and 4, respectively. +The influence of the substitutability degree on the local stability of the equilibrium and related comparative +statics are discussed in Section 5. Numerical simulations are provided in Section 6. Concluding remarks are +given in Section 7. +2 +Model +In our study, we consider a market where two firms compete with each other and produce differentiated +goods. The prices and quantities of the two goods are denoted by pi and qi, respectively, with i = 1, 2. +Furthermore, it is assumed that the market possesses a continuum of identical consumers with a CES utility +function of the form +U(q1, q2) = qα +1 + qα +2 , +2 + +where α (0 < α < 1) is called the substitutability degree between the products. Consumers choose their +consumptions by maximizing the utility subject to the budget constraint +p1q1 + p2q2 = 1. +Consequently, we have the following demand functions (The reader can refer to [3] for the proof). +q1 = pβ +2 +p1 +1 +pβ +1 + pβ +2 +, +q2 = pβ +1 +p2 +1 +pβ +1 + pβ +2 +, +where β = α/(1 − α). Thus, the inverse demands of the two goods are +p1 = +qα−1 +1 +qα +1 + qα +2 +, +p2 = +qα−1 +2 +qα +1 + qα +2 +. +(1) +Accordingly, a decrease in α would make the products less substitutable or more differentiated. +In +particular, if α = 0, the inverse demands become p1 = +1 +2 q1 and p2 = +1 +2 q2 , which means that the two goods +are completely independent. If α = 1, we obtain the inverse demand p1 = p2 = +1 +q1+q2 , which is the same as +the famous isoelastic demand function introduced by Puu [33]. In this case, the prices of the two goods are +equal. That is to say, the two commodities are regarded as indistinguishable or identical by consumers. +The cost functions are assumed to be linear, i.e., +C1(q1) = c1q1, +C2(q2) = c2q2, +where c1 > 0 and c2 > 0. Then the profit of firm i (i = 1, 2) should be +Πi(pi, p−i) = piqi − ciqi = (pi − ci)pβ +−i +pi +1 +pβ +i + pβ +−i +, +(2) +where p−i denotes the price of the commodity produced by the rival. +Furthermore, the gradient adjustment mechanism is formulated as +pi(t + 1) = pi(t) + ki +∂Πi(t) +∂pi(t) , +where ki > 0 controls the adjustment speed of firm i. It is known that +∂Πi +∂pi += +−pβ +−ip1+β +i +β + +� +p2β +−i + pβ +−ipβ +i (1 + β) +� +ci +p2 +i +� +pβ +i + pβ +−i +�2 +. +In short, the model can be described as the following iteration map. +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +p1(t + 1) = p1(t) + k1 +−pβ +2(t)p1+β +1 +(t)β + +� +p2β +2 (t) + pβ +2(t)pβ +1(t) (1 + β) +� +c1 +p2 +1(t) +� +pβ +1(t) + pβ +2(t) +�2 +, +p2(t + 1) = p2(t) + k2 +−pβ +1(t)p1+β +2 +(t)β + +� +p2β +1 (t) + pβ +1(t)pβ +2(t) (1 + β) +� +c2 +p2 +2(t) +� +pβ +2(t) + pβ +1(t) +�2 +. +(3) +This game was first explored by Ahmed et al. [3] only through numerical simulations because no analytical +expressions of the Nash equilibria are available. In this paper, we reconsider this game using methods based +on symbolic computations and explore the influence of the substitutability degree on the local stability of +the equilibrium. One can see that for general β, it is impossible to analyze the equilibrium point of map +(3), because the system will have an exponential parameter. For such systems with exponential parameters, +existing analytical tools are quite limited. Therefore, similar to [3], our study mainly focuses on two specific +cases, namely β = 1 and β = 1/2, which are corresponding to α = 1/2 and α = 1/3, respectively. +3 + +3 +α = 1/2 +If α = 1/2, then β = 1. Hence, map (3) becomes +� +� +� +� +� +� +� +� +� +p1(t + 1) = p1(t) + k1 +−2 p2(t)p2 +1(t) + +� +p2 +2(t) + 2 p2(t)p1(t) +� +c1 +p2 +1(t) (p1(t) + p2(t))2 +, +p2(t + 1) = p2(t) + k2 +−2 p1(t)p2 +2(t) + +� +p2 +1(t) + 2 p1(t)p2(t) +� +c2 +p2 +2(t) (p2(t) + p1(t))2 +. +(4) +From an economic point of view, it is important to identify the number of non-vanishing equilibria +(p1, p2) with p1 > 0 and p2 > 0. In order to compute the equilibrium, we set p1(t + 1) = p1(t) = p1 and +p2(t + 1) = p2(t) = p2. Then the following equations of the equilibrium are acquired. +� +−2p2p2 +1 + +� +p2 +2 + 2p2p1 +� +c1 = 0, +−2p1p2 +2 + +� +p2 +1 + 2p1p2 +� +c2 = 0. +(5) +The triangular decomposition method, which can be viewed as an extension of the Gaussian elimination +method, permits us to analyze the equilibria of non-linear economic models. Both the method of triangu- +lar decomposition and the method of Gaussian elimination can transform a system into triangular forms. +However, the triangular decomposition method is feasible for polynomial systems, while the Gaussian elim- +ination method is just for linear systems. Refer to [5, 19, 23, 36, 37] for more information on triangular +decomposition. Specifically, using the triangular decomposition method, we can decompose the solutions of +system (5) into zeros of the following two triangular polynomial sets. +T11 = [p1, p2] , +T12 = +� +p3 +1 − 4 c1p2 +1 + (4 c2 +1 − 2 c1c2)p1 + 3 c2 +1c2, c1p2 − p2 +1 + 2 c1p1 +� +. +(6) +The zero of T11 is corresponding to the origin (0, 0). Moreover, the non-vanishing equilibria can be +computed from T12. The first polynomial p3 +1 − 4 c1p2 +1 + (4 c2 +1 − 2 c1c2)p1 + 3 c2 +1c2 of T12 is univariate in p1 and +the second polynomial c1p2 − p2 +1 + 2 c1p1 of T12 has degree 1 with respect to p2. Consequently, if we solve p1 +from the first polynomial, then we can substitute the solution of p1 into the second polynomial and easily +obtain p2. As the first polynomial of T12 has degree 3 with respect to p1, we know that there are at most 3 +positive real solutions. Their analytical expressions exist but are quite complicated, though. +This is not an easy task to identify the exact number of positive real solutions if the analytical solutions +of T12 are complicated. However, the first author of this paper and his co-worker [25] proposed an algebraic +algorithm to systematically identify multiplicities of equilibria in semi-algebraic economies without obtaining +the closed-form solutions. We summarize the computational results for map (4) in Proposition 1. Interested +readers can refer to Section 3 of [25] for additional details of the algorithm. +Proposition 1. Let α = 1/2. The iteration map (4) possesses one unique equilibrium (p1, p2) with p1 > 0 +and p2 > 0. +To explore the local stability of the equilibrium, the following Jacobian matrix plays an ambitious role. +J = +�J11 +J12 +J21 +J22 +� +, +where +J11 = p6 +1 + 3 p5 +1p2 + 3 p4 +1p2 +2 + +� +p3 +2 + 2 k1p2 +� +p3 +1 − 6 k1p2p2 +1c1 − 6 k1p2 +2p1c1 − 2 c1k1p3 +2 +p3 +1 (p1 + p2)3 +, +J12 = k1 (2 c1 − p1 + p2) +(p1 + p2)3 +, +J21 = k2 (2 c2 + p1 − p2) +(p1 + p2)3 +, +J22 = p6 +2 + 3 p1p5 +2 + 3 p2 +1p4 +2 + +� +p3 +1 + 2 k2p1 +� +p3 +2 − 6 k2p1p2 +2c2 − 6 k2p2 +1p2c2 − 2 c2k2p3 +1 +p3 +2 (p1 + p2)3 +. +4 + +Then the characteristic polynomial of J is +CP(λ) = λ2 − Tr(J)λ + Det(J), +where Tr(J) = J11 + J22 and Det(J) = J11J22 − J12J21 are the trace and the determinant of J, respectively. +According to the Jury criterion [18], the conditions for the local stability include: +1. CDJ +1 ≡ CP(1) = 1 − Tr(J) + Det(J) > 0, +2. CDJ +2 ≡ CP(−1) = 1 + Tr(J) + Det(J) > 0, +3. CDJ +3 ≡ 1 − Det(J) > 0. +Remark 1. Furthermore, it is known that the discrete dynamic system may undergo a fold, period-doubling, +or Neimark-Sacker bifurcation when the equilibrium loses its stability at CDJ +1 = 0, CDJ +2 = 0, or CDJ +3 = 0, +respectively. +3.1 +The special case of c1 = c2 +If we set c1 = c2 = c in (5), then the triangular decomposition method permits us to transform the +equilibrium equations (5) into the following three triangular sets. +T21 = [p1, p2], +T22 = [p1 − 3 c, p2 − 3 c], +T23 = [p2 +1 − cp1 − c2, p2 + p1 − c]. +The zero of T21 is simply (0, 0). From T23, we obtain two zeros1 +��√ +5 +2 + 1 +2 +� +c, +� +− +√ +5 +2 + 1 +2 +� +c +� +, +�� +− +√ +5 +2 + 1 +2 +� +c, +�√ +5 +2 + 1 +2 +� +c +� +, +which are useless as the component +� +− +√ +5 +2 + 1 +2 +� +c is negative. Therefore, the only non-vanishing equilibrium +is (3 c, 3 c), which can be obtained from T22. +Theorem 1. Let α = 1/2 and c1 = c2 = c. The unique non-vanishing equilibrium (3 c, 3 c) is locally stable +if +c2 > 2 k1 + 2 k2 + +� +4 k2 +1 − 7 k1k2 + 4 k2 +2 +216 +. +The system may undergo a period-doubling bifurcation when +c2 = 2 k1 + 2 k2 + +� +4 k2 +1 − 7 k1k2 + 4 k2 +2 +216 +. +Furthermore, there exist no other bifurcations of the equilibrium. +Proof. Substituting p1 = 3 c and p2 = 3 c into J, we obtain that the Jacobian matrix at (3 c, 3 c) to be +J(3 c, 3 c) = +� +27 c2−k1 +27 c2 +k1 +216 c2 +k2 +216 c2 +27 c2−k2 +27 c2 +� +. +Consequently, +Tr(J) = 54 c2 − k1 − k2 +27 c2 +, +Det(J) = 5184 c4 − 192 c2k1 − 192 c2k2 + 7 k1k2 +5184 c4 +. +1These zeros can also be obtained from T12 in (6) by setting c1 = c2 = c. +5 + +One can verify that the first condition for the local stability is always fulfilled since k1, k2, c > 0 and +CDJ +1 ≡ 1 − Tr(J) + Det(J) = 5 k1k2 +3888 c4 . +The second condition is +CDJ +2 ≡ 1 + Tr(J) + Det(J) = 15552 c4 + (−288 k1 − 288 k2) c2 + 5 k1k2 +3888 c4 +> 0, +which means that +15552 c4 + (−288 k1 − 288 k2) c2 + 5 k1k2 > 0, +i.e., +c2 > 2 k1 + 2 k2 + +� +4 k2 +1 − 7 k1k2 + 4 k2 +2 +216 +or c2 < 2 k1 + 2 k2 − +� +4 k2 +1 − 7 k1k2 + 4 k2 +2 +216 +. +The third condition is +CDJ +3 ≡ 1 − Det(J) = (144 k1 + 144 k2) c2 − 5 k1k2 +3888 c4 +> 0, +which implies that +(144 k1 + 144 k2) c2 − 5 k1k2 > 0, +i.e., +c2 > +5 k1k2 +144 (k1 + k2). +Furthermore, it can be proved that +2 k1 + 2 k2 − +� +4 k2 +1 − 7 k1k2 + 4 k2 +2 +216 +< +5 k1k2 +144 (k1 + k2) < 2 k1 + 2 k2 + +� +4 k2 +1 − 7 k1k2 + 4 k2 +2 +216 +. +Accordingly, the equilibrium is locally stable if +c2 > 2 k1 + 2 k2 + +� +4 k2 +1 − 7 k1k2 + 4 k2 +2 +216 +. +The rest of the proof follows immediately from Remark 1. +Figure 1 depicts two 2-dimensional cross-sections of the stability region reported in Theorem 1. It is +observed that an increase in the marginal cost c or a decrease in the adjustment speeds k1, k2 has an effect +of stabilizing the unique non-vanishing equilibrium. +(a) k2 = 1/10 +(b) c = 1/3 +Figure 1: The 2-dimensional cross-sections of the stability region of the considered model with α = 1/2 and +c1 = c2 = c. The curves of CDJ +2 = 0 and CDJ +3 = 0 are marked in blue and green, respectively. +6 + +3.2 +The general case +If c1 ̸= c2, then the analytical expression of the unique non-vanishing equilibrium would be quite complicated. +Thus, the proof of Theorem 1 can not work since it is impossible to substitute the analytical expression of +the equilibrium into the Jacobian matrix and obtain a neat result. Concerning the bifurcation analysis, we +need to determine the conditions on the parameters that CDJ +1 = 0, CDJ +2 = 0, and CDJ +3 = 0 are satisfied at +the non-vanishing equilibrium. For this purpose, the following notation is required. +Definition 1. Let +A = +m +� +i=0 +ai xi, +B = +l +� +j=0 +bj xj +be two univariate polynomials in x with coefficients ai, bj, and am, bl ̸= 0. The determinant +��������������� +am +am−1 +· · · +a0 +... +... +... +... +am +am−1 +· · · +a0 +bl +bl−1 +· · · +b0 +... +... +... +... +bl +bl−1 +· · · +b0 +��������������� +� +� +� l +� +� +� m +is called the Sylvester resultant (or simply resultant) of A and B with respect to x, and denoted by +res(A, B, x). +The following lemma reveals the main property of the resultant, which can also be found in [30]. +Lemma 1. Let A and B be two univariate polynomials in x. There exist two polynomials F and G in x +such that +FA + GB = res(A, B, x). +Furthermore, A and B have common zeros in the field of complex numbers if and only if res(A, B) = 0. +For a triangular set T = [T1(x), T2(x, y)] and a polynomial H(x, y), we define +res(H, T ) ≡ res(res(H, T2, y), T1(x), x). +By Lemma 1, if T1 = 0 and T2 = 0 (or simply denoted as T = 0), then one knows that H = 0 implies +res(H, T ) = 0, which means res(H, T ) = 0 is a necessary condition for H = 0. Consequently, the following +proposition is acquired. It should be emphasized that Proposition 2 only reports the results for the case +of k1 = k2 because the conditions for k1 ̸= k2 are too long to list in this paper due to space limitations. +However, readers can see that the idea of the proof also works for k1 ̸= k2 and can derive the complete +conditions themself. +Proposition 2. Let α = 1/2 and k1 = k2 = k. The system may undergo a period-doubling bifurcation when +R1 = 0 and a Neimark-Sacker bifurcation when R2 = 0, where R1 and R2 are given in Appendix. +Proof. It should be noted that the resultant is feasible only for polynomials. For CDJ +1 , we consider its +numerator Num(CDJ +1 ). Then one can obtain that +res(Num(CDJ +1 ), T12) = 81 k6c18 +1 c6 +2 (c1 + c2) +� +32c2 +1 + 61c1c2 + 32c2 +2 +� +. +Since c1 > 0, c2 > 0, and k > 0, it is impossible that res(Num(CDJ +1 ), T12) = 0 or CDJ +1 = 0 provided that +T12 = 0. Hence, the equilibrium can not lose its stability through a fold bifurcation. Furthermore, we have +res(Num(CDJ +2 ), T12) = −729 c32 +1 c8 +2(c1 + c2)R1, +res(Num(CDJ +3 ), T12) = 729 k3c32 +1 c8 +2(c1 + c2)R2, +which will vanish only if R1 = 0 and R2 = 0, respectively. +Consequently, the system may undergo a +period-doubling bifurcation when R1 = 0 and a Neimark-Sacker bifurcation when R2 = 0. +7 + +By Proposition 1, there exists only one equilibrium (p1, p2) with p1 > 0 and p2 > 0 although its analytical +expression is complicated. To explore the local stability, we need to determine the signs of CDJ +1 , CDJ +2 , and +CDJ +3 at this equilibrium without using its closed form. It should be noted that CDJ +1 , CDJ +2 , and CDJ +3 are +rational functions. Suppose that +CDJ +i = Num(CDJ +i ) +Den(CDJ +i ) , +where Num(·) and Den(·) denote the numerator and the denominator, respectively. Then the sign of CDJ +i +is the same as that of Num(CDJ +i ) · Den(CDJ +i ) if Den(CDJ +i ) ̸= 0. One could compute that +res(Num(CDJ +1 ) · Den(CDJ +1 ), T12) = −1594323 k6c50 +1 c17 +2 (c1 + c2)6(32 c2 +1 + 61 c1c2 + 32 c2 +2), +res(Num(CDJ +2 ) · Den(CDJ +2 ), T12) = 129140163 c70 +1 c22 +2 (c1 + c2)6R1, +and +res(Num(CDJ +3 ) · Den(CDJ +3 ), T12) = −129140163 k3c70 +1 c22 +2 (c1 + c2)6R2. +We should emphasize that the sign of res(Num(CDJ +i )·Den(CDJ +i ), T12) may not be the same as Num(CDJ +i )· +Den(CDJ +i ) or CDJ +i . However, it is known that res(Num(CDJ +i )·Den(CDJ +i ), T12) involves only the parameters +and its zeros divide the parameter space into several regions. In each region, the sign of CDJ +i is invariant. +Consequently, we just need to select one sample point from each region and identify the sign of CDJ +i at the +selected sample point. The selection of sample points might be extremely complicated in general and could +be automated using, e.g., the PCAD method [11]. +In Table 1, we list all the selected sample points and the corresponding information on whether the +non-vanishing equilibrium is stable, i.e., whether CDJ +1 > 0, CDJ +2 > 0, and CDJ +3 > 0 are simultaneously +satisfied. Moreover, Table 1 displays the signs of R1 and R2 at these sample points. One can observe that +the equilibrium is stable if R1 > 0 and R2 > 0, and vice versa. It should be mentioned that the calculations +involved in Table 1 are exact and rigorous. That is, the computational results provide theoretical foundations +for a systematic analysis of the local stability. Therefore, we acquire the following theorem. +Theorem 2. If k1 = k2 = k, the unique non-vanishing equilibrium (p1, p2) with p1 > 0 and p2 > 0 is locally +stable if R1 > 0 and R2 > 0, where R1 and R2 can be found in Appendix. +Table 1: Selected Sample Points in {(c1, c2, k) | c1 > 0, c2 > 0, k > 0} for α = 1/2 +(c1, c2, k) +stable +R1 +R2 +(c1, c2, k) +stable +R1 +R2 +(1, 1/4, 1) +yes ++ ++ +(1, 5/16, 1) +yes ++ ++ +(1, 1/4, 7) +no +− ++ +(1, 5/16, 10) +no +− ++ +(1, 1/4, 29) +no +− +− +(1, 5/16, 30) +no +− +− +(1, 1/4, 51) +no ++ +− +(1, 5/16, 51) +no ++ +− +(1, 1/2, 1) +yes ++ ++ +([1, 7/8, 1) +yes ++ ++ +(1, 1/2, 18) +no +− ++ +(1, 7/8, 38) +no +− ++ +(1, 1/2, 35) +no +− +− +(1, 7/8, 51) +no +− +− +(1, 1/2, 53) +no ++ +− +(1, 7/8, 65) +no ++ +− +(1, 9/8, 1) +yes ++ ++ +(1, 2, 1) +yes ++ ++ +(1, 9/8, 49) +no +− ++ +(1, 2, 70) +no +− ++ +(1, 9/8, 66) +no +− +− +(1, 2, 140) +no +− +− +(1, 9/8, 83) +no ++ +− +(1, 2, 209) +no ++ +− +(1, 3, 1) +yes ++ ++ +(1, 4, 1) +yes ++ ++ +(1, 3, 91) +no +− ++ +(1, 4, 112) +no +− ++ +(1, 3, 272) +no +− +− +(1, 4, 462) +no +− +− +(1, 3, 453) +no ++ +− +(1, 4, 811) +no ++ +− +8 + +4 +α = 1/3 +If α = 1/3, then β = 1/2. We have the iteration map +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +p1(t + 1) = p1(t) + k1 +−p1(t) +� +p1(t)p2(t) + +� +2 p2(t) + 3 +� +p1(t)p2(t) +� +c1 +2 p2 +1(t) +�� +p1(t) + +� +p2(t) +�2 +, +p2(t + 1) = p2(t) + k2 +−p2(t) +� +p1(t)p2(t) + +� +2 p1(t) + 3 +� +p1(t)p2(t) +� +c2 +2 p2 +2(t) +�� +p1(t) + +� +p2(t) +�2 +. +(7) +By setting p1(t + 1) = p1(t) = p1 and p2(t + 1) = p2(t) = p2, one can obtain the equations of the +equilibrium +� +− p1 +√p1p2 + (2 p2 + 3√p1p2) c1 = 0, +− p2 +√p1p2 + (2 p1 + 3√p1p2) c2 = 0. +Denote √p1 = x and √p2 = y. The above equations become +� +− x3y + (2 y2 + 3 xy)c1 = 0, +− y3x + (2 x2 + 3 xy)c2 = 0. +(8) +Using the triangular decomposition method, we decompose the solutions of system (8) into zeros of the +following two triangular sets. +T31 = [x, y] , +T32 = +� +x8 − 9 c1x6 + 27 c2 +1x4 + (−27 c3 +1 − 12 c2 +1c2)x2 + 20 c3 +1c2, 2 c1y − x3 + 3 c1x +� +. +Evidently, T31 is corresponding to the origin (0, 0). Therefore, the identification of the number of non- +vanishing equilibria can be transformed into the determination of the number of real solutions of the following +semi-algebraic system. +� +� +� +� +� +x8 − 9 c1x6 + 27 c2 +1x4 + (−27 c3 +1 − 12 c2 +1c2)x2 + 20 c3 +1c2 = 0, +2 c1y − x3 + 3 c1x = 0, +x > 0, y > 0. +Using the algebraic approach by Li and Wang [25], we know that the above system has one unique real +solution for any parameter values of c1, c2 > 0, which implies the following proposition. +Proposition 3. Let α = 1/3. The iteration map (7) possesses one unique equilibrium (p1, p2) with p1 > 0 +and p2 > 0. +To investigate the local stability of the equilibrium, we consider the Jacobian matrix +M = +�M11 +M12 +M21 +M22 +� +, +where +M11 = 12 p +9 +2 +1 +√p2 + 4 p +7 +2 +1 p +3 +2 +2 − 15 c1k1p +3 +2 +1 +√p2 − 8 c1k1p +3 +2 +2 +√p1 + 3 k1p +5 +2 +1 +√p2 + 4 p5 +1 + 12 p4 +1p2 − 21 c1k1p1p2 + k1p2 +1p2 +4 p +7 +2 +1 +�√p1 + √p2 +�3 +, +M12 = +k1 +�√p2 p +3 +2 +1 − p2 +1 + c1√p2√p1 + 3 p1c1 +� +4 p2 +1 +�√p1 + √p2 +�3 √p2 +, +M21 = +k2 +�√p1 p +3 +2 +2 + c2√p2√p1 + 3 p2c2 − p2 +2 +� +4 p2 +2 +�√p1 + √p2 +�3 √p1 +, +9 + +M22 = 4 p +3 +2 +1 p +7 +2 +2 + 12 p +9 +2 +2 +√p1 − 8 c2k2p +3 +2 +1 +√p2 − 15 c2k2p +3 +2 +2 +√p1 + 3 k2p +5 +2 +2 +√p1 + 12 p1 p4 +2 + 4 p5 +2 − 21 c2k2p1p2 + k2p1p2 +2 +4p +7 +2 +2 +�√p1 + √p2 +�3 +. +As in Section 3, we denote +CDM +1 ≡ 1 − Tr(M) + Det(M), +CDM +2 ≡ 1 + Tr(M) + Det(M), +CDM +3 ≡ 1 − Det(M). +4.1 +The special case of c1 = c2 +If we set c1 = c2 = c, then the triangular decomposition method permits us to transform the equilibrium +equations (8) into the following triangular sets. +T41 = [x, y], +T42 = [x2 − c, y + x], +T43 = [x2 − 5 c, y − x], +T44 = [x4 − 3 c x2 + 4 c2, 2 cy − x3 + 3 cx]. +Obviously, the zeros of T41 and T42 are economically uninteresting. Moreover, all the roots of x4−3 c x2+ +4 c2 of T44, i.e., +� +2 +√ +7c i + 6 c +2 +, − +� +2 +√ +7c i + 6 c +2 +, +� +−2 +√ +7c i + 6 c +2 +, − +� +−2 +√ +7c i + 6 c +2 +, +are imaginary and also not of our concern. +There exists only one non-vanishing equilibrium (p1, p2) = +(5 c, 5 c), which is corresponding to the branch T43. +Substituting p1 = 5 c and p2 = 5 c into M, we obtain the Jacobian matrix at the equilibrium (5 c, 5 c) to +be +M(5 c, 5 c) = +� +500 c2−3 k1 +500 c2 +k1 +1000 c2 +k2 +1000 c2 +500 c2−3 k2 +500 c2 +� +. +Hence, +Tr(M) = 1000 c2 − 3 k1 − 3k2 +500 c2 +, +Det(M) = 200000 c4 − 1200 c2k1 − 1200 c2k2 + 7 k1k2 +200000 c4 +. +Theorem 3. Let α = 1/3 and c1 = c2 = c. The unique non-vanishing equilibrium (5 c, 5 c) is locally stable +if +c2 > 3 k1 + 3 k2 + +� +9 k2 +1 − 17 k1k2 + 9 k2 +2 +2000 +. +The system may undergo a period-doubling bifurcation when +c2 = 3 k1 + 3 k2 + +� +9 k2 +1 − 17 k1k2 + 9 k2 +2 +2000 +. +Furthermore, there exist no other bifurcations of the equilibrium. +Proof. The first condition for the local stability is always fulfilled since +CDM +1 ≡ 1 − Tr(M) + Det(M) = +7 k1k2 +200000c4 . +The second condition should be +CDM +2 ≡ 1 + Tr(M) + Det(M) = 800000 c4 + (−2400 k1 − 2400 k2) c2 + 7 k1k2 +200000 c4 +> 0, +10 + +which implies that +800000 c4 + (−2400 k1 − 2400 k2) c2 + 7 k1k2 > 0, +i.e., +c2 > 3 k1 + 3 k2 + +� +9 k2 +1 − 17 k1k2 + 9 k2 +2 +2000 +or c2 < 3 k1 + 3 k2 − +� +9 k2 +1 − 17 k1k2 + 9 k2 +2 +2000 +. +The third condition should be +CDM +3 ≡ 1 − Det(M) = (1200 k1 + 1200 k2) c2 − 7 k1k2 +200000 c4 +> 0, +from which we have +(1200 k1 + 1200 k2) c2 − 7 k1k2 > 0, +i.e., +c2 > +7 k1k2 +1200 (k1 + k2). +It can be proved that +3 k1 + 3 k2 − +� +9 k2 +1 − 17 k1k2 + 9 k2 +2 +2000 +< +7 k1k2 +1200 (k1 + k2) < 3 k1 + 3 k2 + +� +9 k2 +1 − 17 k1k2 + 9 k2 +2 +2000 +. +Therefore, the equilibrium is locally stable if +c2 > 3 k1 + 3 k2 + +� +9 k2 +1 − 17 k1k2 + 9 k2 +2 +2000 +. +The rest of the proof follows from Remark 1. +In Figure 2, we show two 2-dimensional cross-sections of the stability region reported in Theorem 3. One +can see that the equilibrium may lose its stability if the adjustment speeds k1, k2 are large enough or the +marginal cost c is small enough. +(a) k2 = 1/10 +(b) c = 1/10 +Figure 2: The 2-dimensional cross-sections of the stability region of the considered model with α = 1/3 and +c1 = c2 = c. The curves of CDM +2 = 0 and CDM +3 = 0 are marked in blue and green, respectively. +4.2 +The general case +As in Section 3.2, we set k1 = k2 = k. We should mention that the method employed in this section also +works for the case of k1 ̸= k2. However, the conditions for k1 ̸= k2 are tedious and not reported in this +section due to space limitations. Interested readers can use our method to compute the complete conditions +themself. The case of c1 = c2 has been explored in Section 4.1, hence we suppose that c1 ̸= c2 in what +follows. The bifurcations are analyzed in the following proposition. +11 + +Proposition 4. Let α = 1/3, k1 = k2 = k and c1 ̸= c2. The iteration map (7) may undergo a period- +doubling bifurcation when R3 = 0 and a Neimark-Sacker bifurcation when R4 = 0, where R3 and R4 are +given in Appendix. +Proof. Computing the resultant of Num(CDM +1 ) with respect to T32, one obtains +res(Num(CDM +1 ), T32) = 879609302220800000 k16c51 +1 c11 +2 (c1 − c2)2 � +2187 c2 +1 − 4031 c1c2 + 2187 c2 +2 +�2 . +It is evident that +2187 c2 +1 − 4031 c1c2 + 2187 c2 +2 = 2187(c1 − c2)2 + 343 c1c2 > 0. +Therefore, res(Num(CDM +1 ), T32) ̸= 0, which means that CDM +1 ̸= 0 at the unique non-vanishing equilibrium. +Hence, there exist no fold bifurcations in map (7). Furthermore, we have +res(Num(CDJ +2 ), T32) = 99035203142830421991929937920000000 c101 +1 +c13 +2 (c1 − c2)2 R2 +3, +and +res(Num(CDJ +3 ), T32) = 99035203142830421991929937920000000 k8c101 +1 +c13 +2 (c1 − c2)10R2 +4. +Consequently, a period-doubling bifurcation may occur when R3 = 0, while a Neimark-Sacker bifurcation +may take place when R4 = 0. +To investigate the local stability, we need to consider Num(CDJ +i ) · Den(CDJ +i ) and compute its resultant +with respect to T32. Then it is obtained that +res(Num(CDJ +1 ) · Den(CDJ +1 ), T32) = +5708990770823839524233143877797980545530986496 · 1020 +· k16c156 +1 +c36 +2 (c1 − c2)12(2187 c2 +1 − 4031 c1c2 + 2187 c2 +2)2, +res((Num(CDJ +2 ) · Den(CDJ +2 ), T32) = +6582018229284824168619876730229402019930943462534319453394436096 · 1024 +· c218 +1 +c42 +2 (c1 − c2)10R2 +3, +res((Num(CDJ +3 ) · Den(CDJ +3 ), T32) = +6582018229284824168619876730229402019930943462534319453394436096 · 1024 +· k8c218 +1 +c42 +2 (c1 − c2)10R2 +4. +These res(Num(CDJ +i )·Den(CDJ +i ), T32) involve only the parameters and their zeros divide the parameter +set {(c1, c2, k) | c1 > 0, c2 > 0, k > 0} into several regions. +In each region, the signs of CDM +1 , CDM +2 , +and CDM +3 +are fixed and can be identified by checking at a selected sample point. In Table 2, we list the +40 selected sample points and the signs of R3, R4 at these sample points. +Moreover, Table 2 provides +the information on whether the non-vanishing equilibrium is stable, i.e., whether the stability conditions +CDM +1 +> 0, CDM +2 +> 0 and CDM +3 +> 0 are satisfied simultaneously. +Interested readers may check the +correctness of Table 2 themselves. Based on a series of computations, we acquire the following theorem. +Theorem 4. Let k1 = k2 = k and c1 ̸= c2. The unique non-vanishing equilibrium of map (7) is locally +stable if one of the following conditions is satisfied: +1. R3 > 0, R4 > 0; +2. R3 < 0, R4 > 0 and A1 > 0, A2 < 0, A3 > 0, +where R3, R4, A1, A2, and A3 can be found in Appendix. +Remark 2. From Table 2, we see that the equilibrium is stable if R3 > 0 and R4 > 0. Hence, R3 > 0, R4 > 0 +is a sufficient condition for the local stability. However, this condition is not necessary. For example, at the +first sample point (1, 1/4, 1/512) listed in Table 2, the equilibrium is locally stable, but one can verify that +R3 < 0 and R4 > 0 at this point. Thus, the second condition of Theorem 4 is needed. +The necessity of the second condition can also be illustrated by Figure 4 (b, d, f), where the regions +defined by the first and second conditions are marked in light grey and dark grey, respectively. By economic +12 + +intuition, we know that for a fixed value of the marginal cost c2, a decrease in the adjustment speed k would +be beneficial to the local stability of the equilibrium. That is to say, the dark grey regions defined by the +second condition would be more likely to be included in the stability regions. +It is noted that A1, A2, and A3 involved in the second condition are contained in the so-called generalized +discriminant list and can be picked out by repeated trials. Concerning the generalized discriminant list, the +reader may refer to [38] for more details. The polynomials A1, A2, and A3 are needed here since the condition +that R3 < 0 and R4 > 0 is not a sufficient condition for the local stability. For example, the model is stable +at (1, 1/4, 1/512), where R3 < 0 and R4 > 0. But, the model is unstable at (1, 1/4, 34), where R3 < 0 and +R4 > 0 are also satisfied. Consequently, additional polynomials are needed to constrict the region defined +by R3 < 0 and R4 > 0 such that the complete stability conditions can be acquired. +Table 2: Selected Sample Points in {(c1, c2, k) | c1 > 0, c2 > 0, k > 0} for α = 1/3 +(c1, c2, k) +stable +R3 +R4 +(c1, c2, k) +stable +R3 +R4 +(1, 1/4, 1/512) +yes +− ++ +(1, 3/8, 1/128) +yes +− ++ +(1, 1/4, 1) +yes ++ ++ +(1, 3/8, 1) +yes ++ ++ +(1, 1/4, 34) +no +− ++ +(1, 3/8, 64) +no +− ++ +(1, 1/4, 153) +no +− +− +(1, 3/8, 175) +no +− +− +(1, 1/4, 273) +no ++ +− +(1, 3/8, 287) +no ++ +− +(1, 5/8, 1/32) +yes +− ++ +(1, 7/8, 1/128) +yes +− ++ +(1, 5/8, 1) +yes ++ ++ +(1, 7/8, 1) +yes ++ ++ +(1, 5/8, 145) +no +− ++ +(1, 7/8, 244) +no +− ++ +(1, 5/8, 231) +no +− +− +(1, 7/8, 302) +no +− +− +(1, 5/8, 317) +no ++ +− +(1, 7/8, 361) +no ++ +− +(1, 5/4, 1/32) +yes +− ++ +(1, 3/2, 1/16) +yes +− ++ +(1, 5/4, 1) +yes ++ ++ +(1, 3/2, 1) +yes ++ ++ +(1, 5/4, 335) +no +− ++ +(1, 3/2, 362) +no +− ++ +(1, 5/4, 436) +no +− +− +(1, 3/2, 544) +no +− +− +(1, 5/4, 538) +no ++ +− +(1, 3/2, 726) +no ++ +− +(1, 2, 1/16) +yes +− ++ +(1, 3, 1/16) +yes +− ++ +(1, 2, 1) +yes ++ ++ +(1, 3, 1) +yes ++ ++ +(1, 2, 403) +no +− ++ +(1, 3, 471) +no +− ++ +(1, 2, 804) +no +− +− +(1, 3, 1503) +no +− +− +(1, 2, 1205) +no ++ +− +(1, 3, 2536) +no ++ +− +5 +Influence of the Substitutability Degree +Firstly, we analyze the influence of the substitutability degree α on the size of the stability region of the +equilibrium. We start by considering the special case of c1 = c2. +Proposition 5. Let c1 = c2. The stability region for α = 1/2 is a proper subset of that for α = 1/3. +Proof. Recall Theorems 1 and 3. We need to prove that +2 k1 + 2 k2 + +� +4 k2 +1 − 7 k1k2 + 4 k2 +2 +216 +> 3 k1 + 3 k2 + +� +9 k2 +1 − 17 k1k2 + 9 k2 +2 +2000 +, +which is equivalent to +� +2 k1 + 2 k2 + +� +4 k2 +1 − 7 k1k2 + 4 k2 +2 +216 +�2 +− +� +3 k1 + 3 k2 + +� +9 k2 +1 − 17 k1k2 + 9 k2 +2 +2000 +�2 +> 0. +The left-hand side of the above inequality can be simplified into +− (4374 k1 + 4374 k2) +� +9 k2 +1 − 17 k1k2 + 9 k2 +2 +2916000000 ++ (250000 k1 + 250000 k2) +� +4 k2 +1 − 7 k1k2 + 4 k2 +2 +2916000000 +13 + ++ 243439 k2 +1 +1458000000 + 61771 k1k2 +2916000000 + 243439 k2 +2 +1458000000. +It is easy to check that +(4374 k1 + 4374 k2) +� +9 k2 +1 − 17 k1k2 + 9 k2 +2 +2916000000 +< (250000 k1 + 250000 k2) +� +4 k2 +1 − 7 k1k2 + 4 k2 +2 +2916000000 +, +which completes the proof. +If c1 ̸= c2, however, the conclusion of the above proposition would be incorrect. For example, if we +assume k1 = k2 = k and take (c1, c2, k) = (261/65536, 1/2, 79/1024), then +R1 = +588713082686404258452596575293972215811486125608829 +6129982163463555433433388108601236734474956488734408704 > 0, +R2 = 108130364702270905134254005155560019343 +340282366920938463463374607431768211456 > 0. +Hence, (261/65536, 1/2, 79/1024) is in the stability region of the model for α = 1/2. But, at the same +parameter point, namely (c1, c2, k) = (261/65536, 1/2, 79/1024), we have +R3 = − 791461358900213183480020700044263844445257635142615074110540187 +26328072917139296674479506920917608079723773850137277813577744384 < 0, +R4 = 526438846625624761986017962528229497389068363385599391 +374144419156711147060143317175368453031918731001856 +> 0, +and +A1 = +44864955 +4294967296 > 0, +A2 = − +842240947483983714275440267 +81129638414606681695789005144064 < 0, +A3 = − 63936547182666560163845458457577 +649037107316853453566312041152512 < 0. +This means that the stability conditions of Theorem 4 for α = 1/3 are not satisfied. +On the other hand, one can also find some points where the model is stable for α = 1/3 but unstable for +α = 1/2. For example, at (c1, c2, k) = (3/8, 1/2, 827/64), we know +R3 = 40079185741889580295152003015 +288230376151711744 +> 0, +R4 = 29339436396656781 +17179869184 +> 0. +Therefore, (3/8, 1/2, 827/64) is in the stability region for α = 1/3. However, +R1 = −24200272602071108539 +17592186044416 +< 0, +R2 = −96467864887 +67108864 +< 0. +That is to say, (3/8, 1/2, 827/64) is an unstable parameter point for α = 1/2. +Figure 3 depicts the 2-dimensional cross-sections of the stability regions for α = 1/2 and α = 1/3. For +comparison purposes, we place the cross-sections for α = 1/2 on the left and those for α = 1/3 on the right. +We set k1 = k2 = k and choose three different values of the parameter k, i.e., k = 1/2, 1, 10, to observe the +effect of variation of k on the size of the stability regions. The curves of R1 = 0 and R3 = 0 are marked in +blue; the curves of R2 = 0 and R3 are marked in green; the curves of A1 = 0, A2 = 0 and A3 = 0 are marked +in red. The stability regions are colored in light grey. From Figure 3, we find that the stability region would +shrink if the firms react or adjust their outputs faster both for α = 1/2 and α = 1/3. Similarly, in Figure +4, we assume that k1 and k2 are identical and choose three different values of c1, i.e., c1 = 1/2, 1, 10. The +regions of R1 > 0, R2 > 0 and those of R3 > 0, R4 > 0 are colored in light grey, while the regions defined by +R3 < 0, R4 > 0, A1 > 0, A2 < 0, A3 > 0 are colored in dark grey. From Figure 4, we observe that increasing +the marginal cost c1 of the first firm could result in the enlargement of the stability region for α = 1/2 and +α = 1/3. +As aforementioned, in the case of c1 ̸= c2 and k1 = k2, it can not be proved that the stability region +for α = 1/3 covers that for α = 1/2. From Figures 3 and 4, however, it seems that the stability region for +α = 1/3 is larger than that for α = 1/2. Consequently, for the Bertrand duopoly model considered in this +paper, we may conclude that increasing the substitutability degree α has an effect of destabilizing the unique +14 + +non-vanishing equilibrium in some sense. In other words, product differentiation might make the considered +model more stable, which is an important finding from an economic point of view. Shy [34] discussed the +traditional view on the degree of product differentiation, i.e., a decrease in product differentiation may result +in an increase in market competition intensity and even a price war among involved firms. The possible +explanation for our finding is that a price war might destabilize the equilibrium of the Bertrand game with +differentiated goods. It should be noted that our conclusion is in contrast with the one by Agliari et al. +[1]. Specifically, Agliari et al. [1] investigated a Cournot duopoly model with differentiated products and +employed the same CES utility function and the same linear cost functions as in our study. +However, +they discovered that a higher degree of product differentiation or a lower degree of substitutability leads to +the destabilization of their model. This contradiction may help reveal the essential difference between the +Bertrand and Cournot oligopolies with differentiated goods. +From an economic point of view, the effects on economic variables such as prices and profits of changing +the substitutability degree are interesting. In the sequel, we focus on the comparative statics in the special +case of identical marginal costs. Let c1 = c2 = c. According to (3), the equilibrium satisfies that +� +− pβ +2p1+β +1 +β + p2β +2 c + (pβ +1pβ +2)(1 + β)c = 0, +− pβ +1p1+β +2 +β + p2β +1 c + (pβ +1pβ +2)(1 + β)c = 0. +(9) +Hence, +−pβ +2p1+β +1 +β + p2β +2 c = −pβ +1p1+β +2 +β + p2β +1 c, +which implies that +(p2β +1 − p2β +2 )c = (p2 − p1)pβ +1pβ +2β. +Without loss of generality, we suppose that p1 ≥ p2. Since c > 0 and β > 0, we know (p2β +1 − p2β +2 )c ≥ 0 and +(p2 − p1)pβ +1pβ +2β ≤ 0, which implies p1 = p2. Plugging p1 = p2 into the first equation of (9), one can solve +p1 = p2 = c(2+β) +β +. Therefore, at the equilibrium q1 = q2 = +β +2 c(2+β). As β = α/(1 − α), we obtain +∂pi +∂α = −2 c +α2 < 0, +∂qi +∂α = +1 +(−2 + α)2 c +> 0. +According to (2), the profits of the two firms would be +Π1 = Π2 = +�c(2 + β) +β +− c +� +β +2 c(2 + β) = +1 +2 + β = 1 + +1 +α − 2. +Hence, for i = 1, 2, +∂Πi +∂α = − +1 +(α − 2)2 < 0. +Recalling the inverse demands (1), for a point (q∗ +1, q∗ +2) on the indifference curve, we define the consumer +surplus of the first product to be +CS1 = +� q∗ +1 +0 +qα−1 +1 +qα +1 + q∗α +2 +dq1 = 1 +α +� q∗ +1 +0 +d(qα +1 + q∗α +2 ) +qα +1 + q∗α +2 += 1 +α ln +� +1 + +�q∗ +1 +q∗ +2 +�α� +. +In the case of c1 = c2, the outputs of the two products are equal at the equilibrium. Therefore, we have +that CS1 = CS2 = 1 +α ln 2. Accordingly, the social welfare is +W = CS1 + CS2 + Π1 + Π2 = 2 +α ln 2 + +2 +α − 2 + 2. +Then it is known that +∂W +∂α = −2 ln 2 +α2 +− +2 +(α − 2)α < 0. +To summarize, in the special case of identical marginal costs, an increase in the substitutability degree +α leads to a stable equilibrium with lower prices, higher supplies, lower profits, and lower welfare. In other +words, the degree of product differentiation is positively related to the prices of the goods, the profits of the +involved companies, and the social welfare, which is consistent with our economic intuition. +15 + +(a) α = 1/2, k = 1/2 +(b) α = 1/3, k = 1/2 +(c) α = 1/2, k = 1 +(d) α = 1/3, k = 1 +(e) α = 1/2, k = 10 +(f) α = 1/3, k = 10 +Figure 3: The 2-dimensional cross-sections of the stability regions for α = 1/2 and α = 1/3 if we set +k1 = k2 = k and fix k = 1/2, 1, 10. The curves of R1 = 0 and R3 = 0 are marked in blue; the curves of +R2 = 0 and R3 are marked in green; the curves of A1 = 0, A2 = 0 and A3 = 0 are marked in red. The +stability regions are colored in light grey. +16 + +(a) α = 1/2, c1 = 1/2 +(b) α = 1/3, c1 = 1/2 +(c) α = 1/2, c1 = 1 +(d) α = 1/3, c1 = 1 +(e) α = 1/2, c1 = 10 +(f) α = 1/3, c1 = 10 +Figure 4: The 2-dimensional cross-sections of the stability regions for α = 1/2 and α = 1/3 if we set +k1 = k2 = k and fix c1 = 1/2, 1, 10. The curves of R1 = 0 and R3 = 0 are marked in blue; the curves of +R2 = 0 and R3 are marked in green; the curves of A1 = 0, A2 = 0 and A3 = 0 are marked in red. The +regions of R1 > 0, R2 > 0 and those of R3 > 0, R4 > 0 are colored in light grey, while the regions defined +by R3 < 0, R4 > 0, A1 > 0, A2 < 0, A3 > 0 are colored in dark grey. +17 + +6 +Numerical Simulations +This section provides numerical simulations to illustrate the complex dynamics of the considered Bertrand +duopoly model. The first purpose of our simulations is to confirm the main conclusion of Section 5 that +increasing the substitutability degree α could destabilize the unique non-vanishing equilibrium. In Figure +5, we depict the 1-dimensional bifurcation diagrams with respect to α, where we fix the other parameters +k1 = k2 = 1, c1 = c2 = 0.2 and set the initial point to be (0.56, 1.06). The bifurcation diagrams against +p1 and p2 are given in Figure 5 (a, c) and (b, d), respectively. It is observed that complex dynamics appear +when α becomes large enough. Specifically, there exists one unique stable equilibrium at first, then a stable +2-cycle orbit, and finally a chaotic set as α varies from 0.1 up to 0.7. +To show the transition clearly, +the 1-dimensional bifurcation diagrams are enlarged for α ∈ (0.55, 0.6) in (c, d). One can see that, when +α = 0.553372, a branching point occurs and the unique fixed point bifurcates into a 2-cycle orbit, which, +however, is not a period-doubling bifurcation point. This 2-cycle orbit loses its stability through a Neimark- +Sacker bifurcation rather than a period-doubling bifurcation at α = 0.577570. +More details can be found in Figure 6, where we plot the phase portraits for k1 = k2 = 1 and c1 = c2 = 0.2 +with the initial point (0.56, 1.06). From Figure 6 (a), we observe that, after the occurrence of a Neimark- +Sacker bifurcation, the 2-cycle orbit (P21(0.464194, 0.607384) and P21(0.607384, 0.464194)) becomes unstable +and bifurcates into two invariant closed orbits when α = 0.58; the unique equilibrium E1(0.492557, 0.492557) +goes to E1new(0.489655, 0.489655) when α = 0.58. +Furthermore, all points on the diagonal line x = y +converge to E1new. The two invariant closed orbits marked in blue are stable and points converge to them +from inside and outside. Figure 6 (b) depicts the phase portrait when α = 0.59 and the other parameters +are set to be the same as (a). From (b), one can discover chaotic attractors with symmetry. The above +observations show that an increase in the substitutability degree α leads to the emergence of instability, +complex dynamics, and even chaos in the considered model. +(a) against p1 +(b) against p2 +(c) against p1 and enlarged for α ∈ (0.55, 0.6) +(d) against p2 and enlarged for α ∈ (0.55, 0.6) +Figure 5: The 1-dimensional bifurcation diagrams with respect to α if we fix k1 = k2 = 1, c1 = c2 = 0.2 and +set the initial point to be (0.56, 1.06). +18 + +4 +pi +3 +2 +1 +0 +0.1 +0.2 +0.3 +0.4 +a0.5 +0.6 +0.77 +9 +54 +3 +2 +1 +0 +0.1 +0.2 +0.3 +0.4 +L +a0.5 +0.6 +0.77 +9 +50.7 +pi +0.6 +NS +BP +0.5 +NS +0.4 +0.3 +0.55 +0.56 +0.57 +0.58 +a0.59 +0.61 +0.9 +0.80.7 +0.6 +NS +BP +0.5 +NS +0.4 +0.3 +0.55 +0.56 +0.57 +0.58 +a0.59 +0.61 +0.9 +0.80.4 +0.45 +0.5 +0.55 +0.6 +0.65 +0.7 +p1 +0.4 +0.45 +0.5 +0.55 +0.6 +0.65 +0.7 +p2 +(a) α = 0.58 +0.4 +0.45 +0.5 +0.55 +0.6 +0.65 +0.7 +0.75 +0.8 +p1 +0.4 +0.45 +0.5 +0.55 +0.6 +0.65 +0.7 +0.75 +0.8 +p2 +(b) α = 0.59 +Figure 6: Phase portraits for k1 = k2 = 1 and c1 = c2 = 0.2 with the initial point (0.56, 1.06). +To illustrate the influence of other parameters, several 2-dimensional bifurcation diagrams are computed +and displayed in the sequel. Figure 7 depicts the 2-dimensional bifurcation diagram of map (4) (α = 1/2) +with respect to k1 and k2 if we fix c1 = 0.3, c2 = 0.4 and set the initial point to be (0.5, 0.8). We detect +periodic orbits with distinct orders and mark the corresponding parameter points in different colors in Figure +7. It should be mentioned that the parameter points where there exist periodic orbits with orders more than +25 are marked in light yellow as well. Two different routes from the unique stable equilibrium to complex +dynamics can be observed. For example, if we fix k2 = 7.5 and change the value of k1 from 0.0 to 10.0, the +dynamics of the system start from one unique stable equilibrium (the dark blue region), then transition to +a stable 2-cycle orbit (the light blue region) and finally to invariant closed orbits as well as chaos (the light +yellow region). This is similar to the route displayed in Figure 5, where the stable 2-cycle loses its stability +through a Neimark-Sacker bifurcation. The other route can be discovered, e.g., if we fix k2 = 2.5 and keep +k1 as a free parameter. Then it is observed that the unique stable equilibrium loses its stability through a +cascade of period-doubling bifurcations. +In Figure 8, we plot the 2-dimensional bifurcation diagram of map (7) (α = 1/3) with respect to k1 +and k2 if fixing c1 = 0.1, c2 = 0.15 and setting the initial point to be (0.6, 0.9). Similar to Figure 7, the +aforementioned two routes from local stability to complex dynamics can also be observed in Figure 8. +The 2-dimensional bifurcation diagrams with respect to c1 and c2 for α = 1/2 and α = 1/3 are displayed +in Figures 9 and 10, respectively. One can see that complicated dynamic phenomena take place if one of +the cost parameters c1, c2 is small enough. Similarly, we find the above two routes to chaotic behavior, i.e., +through a cascade of period-doubling bifurcation and through a Neimark-Sacker bifurcation on a 2-cycle +orbit, which have already been discovered by Ahmed et al. [3]. However, from Figure 9, we also find the +existence of a Neimark-Sacker bifurcation directly on the unique equilibrium, which is a new result that +has not been observed by Ahmed et al. [3] yet. Specifically, Figure 9 shows that, if we fix c1 = 0.9 and +decrease the value of c2 from 1.0 to 0.0, the dynamics of the system directly transition from the unique +stable equilibrium (the dark blue region) to invariant closed orbits (the light yellow region). In this case, +the behavior of the market suddenly changes from an ordered state to a disordered state at some critical +point, which can hardly be learned by even rational players. +7 +Concluding Remarks +In this paper, we investigated the local stability, bifurcations, and comparative statics of a dynamic Bertrand +duopoly game with differentiated products. This duopoly is assumed to possess two boundedly rational +players adopting a gradient adjustment mechanism and a continuum of identical consumers with a CES +utility function. Moreover, the cost functions are supposed to be linear. It should be mentioned that the +nonlinearity of the resulting demand function derived from the underlying utility permits us to extend the +applications of Bertrand games to more realistic economies, compared to the widely used Bertrand models +with linear demands. +The considered game was first explored by Ahmed et al. [3], where only numerical simulations are +19 + +Figure 7: The 2-dimensional bifurcation diagram of map (4) (α = 1/2) with respect to k1 and k2 if we fix +c1 = 0.3, c2 = 0.4 and set the initial point to be (0.5, 0.8). +Figure 8: The 2-dimensional bifurcation diagram of map (7) (α = 1/3) with respect to k1 and k2 if we fix +c1 = 0.1, c2 = 0.15 and set the initial point to be (0.6, 0.9). +20 + +10.00 +25 +24 +23 +22 +21 +20 +7.50 +19 +18 +17 +16 +15 +14 +5.00 +13 +12 +11 +10 +8 +2.50 +6 +5 +3 +2 +0.00 +0.00 +2.50 +5.00 +7.50 +10.00 +k16.00 +25 +24 +23 + 22 +21 +20 +4.50 +19 +18 +17 +16 +15 +14 +3.00 +13 +12 +11 +10 +9 +1.50 +6 +5 +3 +2 +0.00 +0.00 +1.50 +3.00 +4.50 +6.00 +k1Figure 9: The 2-dimensional bifurcation diagram of map (4) (α = 1/2) with respect to c1 and c2 if we fix +k1 = 6, k2 = 12 and set the initial point to be (0.5, 0.8). +Figure 10: The 2-dimensional bifurcation diagram of map (7) (α = 1/3) with respect to c1 and c2 if we fix +k1 = 0.3, k2 = 0.6 and set initial point to be (0.6, 0.9). +21 + +1.00 +2. +25 +24 +23 + 22 +21 +20 +0.75. +19 +18 +17 +16 +15 +14 +0.50 +13 +12 +11 +10 +9 +8 +0.25. +6 +5 +4 +3 +2 +0.00 +0.00 +0.25 +0.50 +0.75 +1.00 +C10.10 +24 +25 + 24 +23 + 22 +21 +20 +0.08 +19 +18 +17 +16 +15 +14 +0.05 +13 +12 +11 +10 +9 +0.03 +5 +4 +3 +2 +0.00 +0.00 +0.03 +0.05 +0.08 +0.10 +C1employed to investigate the dynamic behavior and it was observed that the Nash equilibrium loses its +stability through a period-doubling bifurcation as the speed of adjustment increases. In our study, however, +we re-investigated this game using several tools based on symbolic computations such as the triangular +decomposition method (refer to, e.g., [23]) and the PCAD method (refer to, e.g., [11]). +The results of +symbolic computations are exact, and thus provide theoretical foundations for the systematic analysis of +economic models. +For simplicity, our work mainly focused on two specific degrees of product substitutability, namely +α = 1/2 and α = 1/3. In both cases, we proved the uniqueness of the non-vanishing equilibrium using the +algebraic approach of detecting the multiplicity of equilibria proposed by the first author and his co-worker +[25]. We introduce several tools based on symbolic computations and used them to obtain the rigorous +conditions for the local stability of the unique non-vanishing equilibrium for the first time. In the special +case that the two firms have identical marginal costs, we proved that the model can lose its stability only +through a period-doubling bifurcation. From an economic point of view, the most interesting finding was +that an increase in the substitutability degree or a decrease in the product differentiation leads to the +destabilization of the Bertrand model. This is because a price war, which might destabilize the equilibrium, +can take place if the substitutability degree is large enough. +We should mention that our finding is in +contrast with that by Agliari et al. [1] and that by Fanti and Gori [14]. This contradiction contributes to the +literature on the connection between Cournot and Bertrand oligopolies and may help reveal the essential +difference between them. Moreover, we conducted the comparative statics in the special case of identical +marginal costs. The resulting conclusion was that lower degrees of product differentiation mean lower prices, +higher supplies, lower profits, and lower social welfare, which is consistent with our economic intuition. +Numerical simulations were provided in the end, through which complex dynamics such as periodic +orbits and chaos can be observed. The simulations confirmed that an increase in the substitutability degree +α leads to the emergence of instability, complex dynamics, and even chaos in the considered model. Two- +dimensional bifurcation diagrams were also provided to show different possible routes to chaotic behavior, +e.g., through a cascade of period-doubling bifurcation and through a Neimark-Sacker bifurcation on a 2-cycle +orbit. Furthermore, we discovered the existence of a Neimark-Sacker bifurcation directly on the equilibrium, +which is a new finding and has not yet been discovered by Ahmed et al. [3]. +Appendix +R1 = 15552 c10 +1 c6 +2 + 62208 c9 +1c7 +2 + 93312 c8 +1c8 +2 + 62208 c7 +1c9 +2 + 15552 c6 +1c10 +2 + 73728 c11 +1 c3 +2k + 327168 c10 +1 c4 +2k ++ 576576 c9 +1c5 +2k + 541440 c8 +1c6 +2k + 436608 c7 +1c7 +2k + 541440 c6 +1c8 +2k + 576576 c5 +1c9 +2k + 327168 c4 +1c10 +2 k ++ 73728 c3 +1c11 +2 k + 32768 c11 +1 c2 k2 + 94208 c10 +1 c2 +2k2 + 284160 c9 +1c3 +2k2 + 1163712 c8 +1c4 +2k2 + 2855520 c7 +1c5 +2k2 ++ 3825168 c6 +1c6 +2k2 + 2855520 c5 +1c7 +2k2 + 1163712 c4 +1c8 +2k2 + 284160 c3 +1c9 +2k2 + 94208 c2 +1c10 +2 k2 + 32768 c1c11 +2 k2 ++ 77824 c9 +1c2 k3 + 359936 c8 +1c2 +2k3 + 644608 c7 +1c3 +2k3 + 610976 c6 +1c4 +2k3 + 494368 c5 +1c5 +2k3 + 610976 c4 +1c6 +2k3 ++ 644608 c3 +1c7 +2k3 + 359936 c2 +1c8 +2k3 + 77824 c1c9 +2k3 − 4096c8 +1k4 − 12288 c7 +1c2 k4 + 4544 c6 +1c2 +2k4 ++ 70360 c5 +1c3 +2k4 + 114600 c4 +1c4 +2k4 + 70360 c3 +1c5 +2k4 + 4544 c2 +1c6 +2k4 − 12288 c1c7 +2k4 − 4096 c8 +2k4 +− 1024 c5 +1c2 k5 − 3232 c4 +1c2 +2k5 − 4488 c3 +1c3 +2k5 − 3232 c2 +1c4 +2k5 − 1024 c1c5 +2k5 + 32 c3 +1c2 k6 + 61 c2 +1c2 +2k6 ++ 32 c1c3 +2k6, +R2 = 1152 c8 +1c2 +2 + 5832 c7 +1c3 +2 + 12960 c6 +1c4 +2 + 16560 c5 +1c5 +2 + 12960 c4 +1c6 +2 + 5832 c3 +1c7 +2 + 1152 c2 +1c8 +2 + 1024 c8 +1k ++ 3584 c7 +1c2 k + 5920 c6 +1c2 +2k + 6224 c5 +1c3 +2k + 5836 c4 +1c4 +2k + 6224 c3 +1c5 +2k + 5920 c2 +1c6 +2k + 3584 c1c7 +2k ++ 1024 c8 +2k + 512 c5 +1c2 k2 + 1616 c4 +1c2 +2k2 + 2244 c3 +1c3 +2k2 + 1616 c2 +1c4 +2k2 + 512 c1c5 +2k2 − 32 c3 +1c2 k3 +− 61 c2 +1c2 +2k3 − 32 c1c3 +2k3, +R3 = − 209715200000 c12 +1 c8 +2 + 838860800000 c11 +1 c9 +2 − 1258291200000 c10 +1 c10 +2 + 838860800000 c9 +1c11 +2 +− 209715200000 c8 +1c12 +2 + 1160950579200 c13 +1 c5 +2k − 5170397184000 c12 +1 c6 +2k + 9284105011200 c11 +1 c7 +2k +− 9178054656000 c10 +1 c8 +2k + 7806792499200 c9 +1c9 +2k − 9178054656000 c8 +1c10 +2 k + 9284105011200 c7 +1c11 +2 k +− 5170397184000 c6 +1c12 +2 k + 1160950579200 c5 +1c13 +2 k + 626913312768 c13 +1 c3 +2k2 − 1827529703424 c12 +1 c4 +2k2 ++ 6377496477696 c11 +1 c5 +2k2 − 24562717922304 c10 +1 c6 +2k2 + 56911413825536 c9 +1c7 +2k2 +22 + +− 74841436780544 c8 +1c8 +2k2 + 56911413825536 c7 +1c9 +2k2 − 24562717922304 c6 +1c10 +2 k2 ++ 6377496477696 c5 +1c11 +2 k2 − 1827529703424 c4 +1c12 +2 k2 + 626913312768 c3 +1c13 +2 k2 − 117546246144 c12 +1 c2 +2k3 ++ 2268751389696 c11 +1 c3 +2k3 − 8446241806848 c10 +1 c4 +2k3 + 13848228389376 c9 +1c5 +2k3 − 12871123435008 c8 +1c6 +2k3 ++ 10570707526656 c7 +1c7 +2k3 − 12871123435008 c6 +1c8 +2k3 + 13848228389376 c5 +1c9 +2k3 +− 8446241806848 c4 +1c10 +2 k3 + 2268751389696 c3 +1c11 +2 k3 − 117546246144 c2 +1c12 +2 k3 + 7346640384 c11 +1 c2 k4 ++ 23872802112 c10 +1 c2 +2k4 − 79144786368 c9 +1c3 +2k4 − 389232360000 c8 +1c4 +2k4 + 1762366805056 c7 +1c5 +2k4 +− 2639431381760 c6 +1c6 +2k4 + 1762366805056 c5 +1c7 +2k4 − 389232360000 c4 +1c8 +2k4 − 79144786368 c3 +1c9 +2k4 ++ 23872802112 c2 +1c10 +2 k4 + 7346640384 c1c11 +2 k4 − 153055008 c10 +1 k5 + 444048480 c9 +1c2 k5 +− 2281361760 c8 +1c2 +2k5 − 6359031360 c7 +1c3 +2k5 + 33853070112 c6 +1c4 +2k5 − 51945109632 c5 +1c5 +2k5 ++ 33853070112 c4 +1c6 +2k5 − 6359031360 c3 +1c7 +2k5 − 2281361760 c2 +1c8 +2k5 + 444048480 c1c9 +2k5 +− 153055008 c10 +2 k5 + 36636624 c7 +1c2 k6 − 65578896 c6 +1c2 +2k6 + 239834412 c5 +1c3 +2k6 − 377249916 c4 +1c4 +2k6 ++ 239834412 c3 +1c5 +2k6 − 65578896 c2 +1c6 +2k6 + 36636624c1c7 +2k6 − 669222 c5 +1c2 k7 + 1023534 c4 +1c2 +2k7 +− 951468 c3 +1c3 +2k7 + 1023534 c2 +1c4 +2k7 − 669222 c1c5 +2k7 + 2187 c3 +1c2 k8 − 4031 c2 +1c2 +2k8 + 2187 c1c3 +2k8, +R4 = 17714700 c10 +1 c2 +2 − 84798900 c9 +1c3 +2 + 166819500 c8 +1c4 +2 − 187523100 c7 +1c5 +2 + 175575600 c6 +1c6 +2 − 187523100 c5 +1c7 +2 ++ 166819500 c4 +1c8 +2 − 84798900 c3 +1c9 +2 + 17714700 c2 +1c10 +2 + 19131876 c10 +1 k − 55506060 c9 +1c2 k ++ 70441812 c8 +1c2 +2k − 70683840 c7 +1c3 +2k + 106503012 c6 +1c4 +2k − 136123200 c5 +1c5 +2k + 106503012 c4 +1c6 +2k +− 70683840 c3 +1c7 +2k + 70441812 c2 +1c8 +2k − 55506060 c1c9 +2k + 19131876 c10 +2 k − 9159156 c7 +1c2 k2 ++ 23480604 c6 +1c2 +2k2 − 24625107 c5 +1c3 +2k2 + 19286271 c4 +1c4 +2k2 − 24625107 c3 +1c5 +2k2 + 23480604 c2 +1c6 +2k2 +− 9159156 c1c7 +2k2 + 334611 c5 +1c2 k3 − 511767 c4 +1c2 +2k3 + 475734 c3 +1c3 +2k3 − 511767 c2 +1c4 +2k3 + 334611 c1c5 +2k3 +− 2187 c3 +1c2 k4 + 4031 c2 +1c2 +2k4 − 2187 c1c3 +2k4, +A1 = 243 c2 +1 + 352 c1c2 − 9 k, +A2 = − 8000 c5 +1c3 +2 + 19683 c6 +1k − 17496 c5 +1c2 k + 3024 c4 +1c2 +2k + 1728 c3 +1c3 +2k − 2187 c4 +1k2 + 3564 c3 +1c2 k2 +− 432 c2 +1c2 +2k2 + 81 c2 +1k3 + 36 c1c2 k3 − k4, +A3 = 12754584 c7 +1 − 12171384 c6 +1c2 + 3708504 c5 +1c2 +2 + 84096 c4 +1c3 +2 + 2519424 c3 +1c4 +2 − 72171 c5 +1k − 3576744 c4 +1c2 k ++ 5126856 c3 +1c2 +2k − 629856 c2 +1c3 +2k − 25272 c3 +1k2 + 98966 c2 +1c2 k2 + 52488 c1c2 +2k2 + 387 c1 k3 − 1458 c2 k3. +Acknowledgements +The authors wish to thank Dr. Li Su for the beneficial discussions. The authors are grateful to the anonymous +referees for their helpful comments. +This work has been supported by Philosophy and Social Science Foundation of Guangdong (Grant No. +GD21CLJ01), Natural Science Foundation of Anhui Province (Grant No. 2008085QA09), University Natural +Science Research Project of Anhui Province (Grant No. KJ2021A0482), Major Research and Cultivation +Project of Dongguan City University (Grant No. 2021YZDYB04Z). +References +[1] A. Agliari, A. Naimzada, and N. Pecora. Nonlinear dynamics of a Cournot duopoly game with differ- +entiated products. Applied Mathematics and Computation, 281:1–15, 2016. +[2] E. Ahmed, H. Agiza, and S. Hassan. On modifications of Puu’s dynamical duopoly. Chaos, Solitons & +Fractals, 11(7):1025–1028, 2000. +[3] E. Ahmed, A. Elsadany, and T. Puu. On Bertrand duopoly game with differentiated goods. Applied +Mathematics and Computation, 251:169–179, 2015. +[4] S. Askar. 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Tramontana. +Dynamic properties of a Cournot–Bertrand duopoly game with +differentiated products. Economic Modelling, 29(4):1436–1439, 2012. +[33] T. Puu. Chaos in duopoly pricing. Chaos, Solitons & Fractals, 1(6):573–581, 1991. +[34] O. Shy. Industrial Organization: Theory and Applications. MIT Press, Cambridge, 1995. +[35] N. Singh and X. Vives. Price and Quantity Competition in a Differentiated Duopoly. The RAND +Journal of Economics, 15(4):546–554, 1984. +[36] D. Wang. +Computing triangular systems and regular systems. +Journal of Symbolic Computation, +30(2):221–236, 2000. +[37] W.-T. Wu. +Basic principles of mechanical theorem proving in elementary geometries. +Journal of +Automated Reasoning, 2(3):221–252, 1986. +[38] L. Yang, X. Hou, and B. Xia. A complete algorithm for automated discovering of a class of inequality- +type theorems. Science in China Series F, 44:33–49, 2001. +[39] A. Zhang and Y. Zhang. Stability of a Cournot-Nash equilibrium: the multiproduct case. 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Chaos, +Solitons & Fractals, 39(5):2048–2055, 2009. +25 + diff --git a/9NAzT4oBgHgl3EQfFPpp/content/tmp_files/load_file.txt b/9NAzT4oBgHgl3EQfFPpp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2b09af2690d72c5c114a4013c0101d3c4c30f3dc --- /dev/null +++ b/9NAzT4oBgHgl3EQfFPpp/content/tmp_files/load_file.txt @@ -0,0 +1,1389 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf,len=1388 +page_content='A Bertrand duopoly game with differentiated products reconsidered Xiaoliang Lia and Bo Li∗b aSchool of Digital Economics, Dongguan City University, Dongguan 523419, China bSchool of Finance, Anhui University of Finance and Economics, Bengbu 233030, China Abstract In this paper, we explore a dynamic Bertrand duopoly game with differentiated products, where firms are boundedly rational and consumers are assumed to possess an underlying CES utility function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' We mainly focus on two distinct degrees of product substitutability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Several tools based on symbolic computations such as the triangular decomposition method and the PCAD method are employed in the analytical investigation of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The uniqueness of the non-vanishing equilibrium is proved and rigorous conditions for the local stability of this equilibrium are established for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Most importantly, we find that increasing the substitutability degree or decreasing the product differentiation has an effect of destabilization for our Bertrand model, which is in contrast with the relative conclusions for the Cournot models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' This finding could be conducive to the revelation of the essential difference between dynamic Cournot and Bertrand oligopolies with differentiated goods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' In the special case of identical marginal costs, we derive that lower degrees of product differentiation mean lower prices, higher supplies, lower profits, and lower social welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Furthermore, complex dynamics such as periodic orbits and chaos are reported through our numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Keywords: Bertrand duopoly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' differentiated product;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' symbolic computation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' local stability 1 Introduction It is well known that Cournot [12] developed the first formal theory of oligopoly, which is a market supplied by only a few firms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' In Cournot’s framework, firms are supposed to make decisions on their quantities of outputs and have perfect information on their rivals’ strategic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' In the strand of Cournot oligopoly models, the market demand function is usually supposed to be linear for simplicity by many economists (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=', Fisher [16], McManus and Quandt[29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' In the real world, however, a non-linear demand is more likely to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Puu [33] investigated a Cournot duopoly game under an isoelastic market demand, where the price is simply the reciprocal of the total supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Afterward, fruitful contributions including [2, 4, 7, 9, 10, 13, 20, 21, 22, 24, 28, 31], were made in the literature on Cournot games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Related to our study, Zhang and Zhang [39] considered a Cournot game in which each firm produces multiple products and sells them in multiple markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' They obtained sufficient and necessary conditions for the local stability of the Cournot-Nash equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Several decades later after Cournot’s seminal work, Bertrand [6] proposed a different framework to describe oligopolistic competition, where prices rather than quantities are the strategic variables of the competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Singh and Vives [35] analyzed the duality of prices and quantities, and found that Cournot (Bertrand) competition with substitutes is the dual of Bertrand (Cournot) competition with complements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' L´opez and Naylor [26] compared Cournot and Bertrand equilibria in a downstream differentiated duopoly, and proved that the classic conclusion that profits under Cournot equilibrium exceed those under Bertrand competition could be reversible in the case of imperfect substitutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' [40] considered a Bertrand model formulated under a linear inverse demand, and obtained the existence and stability of the equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Different from [40], Fanti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' [15] developed a model with sound microeconomic foundations that deter- mine the demand for differentiated products, and showed that synchronized dynamics and intermittency phenomena may appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Naimzada and Tramontana [32] also considered a Cournot-Bertrand duopoly model with product differentiation and emphasized the role of best response dynamics and an adaptive adjustment mechanism for stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Brianzoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' [8] assumed quadratic costs in the study of the Bertrand duopoly ∗Corresponding author: libomaths@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='com 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='01007v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='TH] 3 Jan 2023 game with horizontal product differentiation and discovered synchronized dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Moreover, Ma and Guo [27] studied the impacts of information on the dynamical Bertrand game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' They showed that there exists a fixed point independent of the amount of information for a triopoly, and the stable region of adjustment parameter increases with the amount of information for a duopoly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' In all the aforementioned Bertrand games, the inverse demand function is supposed to be linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Instead, Gori and Sodini [17] explored the local and global dynamics of a Bertrand duopoly with a nonlinear demand and horizontal product differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Furthermore, Ahmed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' [3] proposed a dynamic Bertrand duopoly game with differentiated products, where firms are boundedly rational and consumers are assumed to possess an underlying CES utility function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' They only employed numerical simulations to investigate the dynamic behavior of their model because the closed form of the equilibrium is extremely difficult to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' They observed that the Nash equilibrium loses its stability through a period-doubling bifurcation as the speed of adjustment increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Motivated by [3], Agliari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' [1] investigated a Cournot duopoly game with differentiated goods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' We should mention that Agliari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' [1] used the same CES utility function as [3] to derive the demand function of the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' They discovered that a low degree of product substitutability or a higher degree of product differentiation may destabilize the Cournot game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' This finding is in accordance with that of Fanti and Gori [14], where the authors introduced a Cournot duopoly with a linear demand and heterogeneous players to study the influence of product differentiation on stability and found that a higher degree of product differentiation may destabilize the market equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' In this paper, we re-study the Bertrand duopoly game of Ahmed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' [3] using several tools based on symbolic computations such as the triangular decomposition method (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=', [23]) and the PCAD method (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=', [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' It is worth noting that the results of symbolic computations are exact, and thus can provide theoretical foundations for the systematic analysis of economic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' We analytically investigate the local stability and bifurcations of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' By using several tools based on symbolic computations, the uniqueness of the non-vanishing equilibrium is proved and the rigorous conditions for the local stability of this equilibrium are obtained for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' In the special case that the two companies have identical marginal costs, we prove that the model can lose its stability only through a period-doubling bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The most important finding is that increasing the substitutability degree or decreasing the product differentiation has an effect of destabilizing the unique non-vanishing equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' A possible explanation is that a decrease in product differentiation may result in an increase in market competition intensity and even a price war, which could lead to the destabilization of the equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' It should be noted that our finding is in contrast with the relative conclusions by Agliari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' [1] and by Fanti and Gori [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' This contradiction contributes to the literature on the connection between Cournot and Bertrand oligopolies and may help reveal the essential difference between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' In the special case of identical marginal costs, we derive the fact that lower degrees of product differentiation can lead to lower prices, higher supplies, lower profits, and lower social welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' This fact is in line with our economic intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Complex dynamics such as periodic orbits and chaos can be observed through our numerical simulations, which also confirm that an increase in the substitutability degree leads to the emergence of instability in the considered model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Furthermore, we discover the existence of a Neimark-Sacker bifurcation directly on the equilibrium, which is a new finding and has not yet been discovered by Ahmed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' [3] The rest of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' In Section 2, we revisit the construction of the Bertrand duopoly game investigated in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' We analytically explore the stability and bifurcations of this model for two different substitutability degrees, namely α = 1/2 and α = 1/3, in Sections 3 and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The influence of the substitutability degree on the local stability of the equilibrium and related comparative statics are discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Numerical simulations are provided in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Concluding remarks are given in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 2 Model In our study, we consider a market where two firms compete with each other and produce differentiated goods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The prices and quantities of the two goods are denoted by pi and qi, respectively, with i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Furthermore, it is assumed that the market possesses a continuum of identical consumers with a CES utility function of the form U(q1, q2) = qα 1 + qα 2 , 2 where α (0 < α < 1) is called the substitutability degree between the products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Consumers choose their consumptions by maximizing the utility subject to the budget constraint p1q1 + p2q2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Consequently, we have the following demand functions (The reader can refer to [3] for the proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' q1 = pβ 2 p1 1 pβ 1 + pβ 2 , q2 = pβ 1 p2 1 pβ 1 + pβ 2 , where β = α/(1 − α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Thus, the inverse demands of the two goods are p1 = qα−1 1 qα 1 + qα 2 , p2 = qα−1 2 qα 1 + qα 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' (1) Accordingly, a decrease in α would make the products less substitutable or more differentiated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' In particular, if α = 0, the inverse demands become p1 = 1 2 q1 and p2 = 1 2 q2 , which means that the two goods are completely independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' If α = 1, we obtain the inverse demand p1 = p2 = 1 q1+q2 , which is the same as the famous isoelastic demand function introduced by Puu [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' In this case, the prices of the two goods are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' That is to say, the two commodities are regarded as indistinguishable or identical by consumers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The cost functions are assumed to be linear, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=', C1(q1) = c1q1, C2(q2) = c2q2, where c1 > 0 and c2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Then the profit of firm i (i = 1, 2) should be Πi(pi, p−i) = piqi − ciqi = (pi − ci)pβ −i pi 1 pβ i + pβ −i , (2) where p−i denotes the price of the commodity produced by the rival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Furthermore, the gradient adjustment mechanism is formulated as pi(t + 1) = pi(t) + ki ∂Πi(t) ∂pi(t) , where ki > 0 controls the adjustment speed of firm i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' It is known that ∂Πi ∂pi = −pβ −ip1+β i β + � p2β −i + pβ −ipβ i (1 + β) � ci p2 i � pβ i + pβ −i �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' In short, the model can be described as the following iteration map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' � � � � � � � � � � � � � � � � � � � p1(t + 1) = p1(t) + k1 −pβ 2(t)p1+β 1 (t)β + � p2β 2 (t) + pβ 2(t)pβ 1(t) (1 + β) � c1 p2 1(t) � pβ 1(t) + pβ 2(t) �2 , p2(t + 1) = p2(t) + k2 −pβ 1(t)p1+β 2 (t)β + � p2β 1 (t) + pβ 1(t)pβ 2(t) (1 + β) � c2 p2 2(t) � pβ 2(t) + pβ 1(t) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' (3) This game was first explored by Ahmed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' [3] only through numerical simulations because no analytical expressions of the Nash equilibria are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' In this paper, we reconsider this game using methods based on symbolic computations and explore the influence of the substitutability degree on the local stability of the equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' One can see that for general β, it is impossible to analyze the equilibrium point of map (3), because the system will have an exponential parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' For such systems with exponential parameters, existing analytical tools are quite limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Therefore, similar to [3], our study mainly focuses on two specific cases, namely β = 1 and β = 1/2, which are corresponding to α = 1/2 and α = 1/3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 3 3 α = 1/2 If α = 1/2, then β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Hence, map (3) becomes � � � � � � � � � p1(t + 1) = p1(t) + k1 −2 p2(t)p2 1(t) + � p2 2(t) + 2 p2(t)p1(t) � c1 p2 1(t) (p1(t) + p2(t))2 , p2(t + 1) = p2(t) + k2 −2 p1(t)p2 2(t) + � p2 1(t) + 2 p1(t)p2(t) � c2 p2 2(t) (p2(t) + p1(t))2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' (4) From an economic point of view, it is important to identify the number of non-vanishing equilibria (p1, p2) with p1 > 0 and p2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' In order to compute the equilibrium, we set p1(t + 1) = p1(t) = p1 and p2(t + 1) = p2(t) = p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Then the following equations of the equilibrium are acquired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' � −2p2p2 1 + � p2 2 + 2p2p1 � c1 = 0, −2p1p2 2 + � p2 1 + 2p1p2 � c2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' (5) The triangular decomposition method, which can be viewed as an extension of the Gaussian elimination method, permits us to analyze the equilibria of non-linear economic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Both the method of triangu- lar decomposition and the method of Gaussian elimination can transform a system into triangular forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' However, the triangular decomposition method is feasible for polynomial systems, while the Gaussian elim- ination method is just for linear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Refer to [5, 19, 23, 36, 37] for more information on triangular decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Specifically, using the triangular decomposition method, we can decompose the solutions of system (5) into zeros of the following two triangular polynomial sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' T11 = [p1, p2] , T12 = � p3 1 − 4 c1p2 1 + (4 c2 1 − 2 c1c2)p1 + 3 c2 1c2, c1p2 − p2 1 + 2 c1p1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' (6) The zero of T11 is corresponding to the origin (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Moreover, the non-vanishing equilibria can be computed from T12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The first polynomial p3 1 − 4 c1p2 1 + (4 c2 1 − 2 c1c2)p1 + 3 c2 1c2 of T12 is univariate in p1 and the second polynomial c1p2 − p2 1 + 2 c1p1 of T12 has degree 1 with respect to p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Consequently, if we solve p1 from the first polynomial, then we can substitute the solution of p1 into the second polynomial and easily obtain p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' As the first polynomial of T12 has degree 3 with respect to p1, we know that there are at most 3 positive real solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Their analytical expressions exist but are quite complicated, though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' This is not an easy task to identify the exact number of positive real solutions if the analytical solutions of T12 are complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' However, the first author of this paper and his co-worker [25] proposed an algebraic algorithm to systematically identify multiplicities of equilibria in semi-algebraic economies without obtaining the closed-form solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' We summarize the computational results for map (4) in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Interested readers can refer to Section 3 of [25] for additional details of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Let α = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The iteration map (4) possesses one unique equilibrium (p1, p2) with p1 > 0 and p2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' To explore the local stability of the equilibrium, the following Jacobian matrix plays an ambitious role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' J = �J11 J12 J21 J22 � , where J11 = p6 1 + 3 p5 1p2 + 3 p4 1p2 2 + � p3 2 + 2 k1p2 � p3 1 − 6 k1p2p2 1c1 − 6 k1p2 2p1c1 − 2 c1k1p3 2 p3 1 (p1 + p2)3 , J12 = k1 (2 c1 − p1 + p2) (p1 + p2)3 , J21 = k2 (2 c2 + p1 − p2) (p1 + p2)3 , J22 = p6 2 + 3 p1p5 2 + 3 p2 1p4 2 + � p3 1 + 2 k2p1 � p3 2 − 6 k2p1p2 2c2 − 6 k2p2 1p2c2 − 2 c2k2p3 1 p3 2 (p1 + p2)3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 4 Then the characteristic polynomial of J is CP(λ) = λ2 − Tr(J)λ + Det(J), where Tr(J) = J11 + J22 and Det(J) = J11J22 − J12J21 are the trace and the determinant of J, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' According to the Jury criterion [18], the conditions for the local stability include: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' CDJ 1 ≡ CP(1) = 1 − Tr(J) + Det(J) > 0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' CDJ 2 ≡ CP(−1) = 1 + Tr(J) + Det(J) > 0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' CDJ 3 ≡ 1 − Det(J) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Furthermore, it is known that the discrete dynamic system may undergo a fold, period-doubling, or Neimark-Sacker bifurcation when the equilibrium loses its stability at CDJ 1 = 0, CDJ 2 = 0, or CDJ 3 = 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1 The special case of c1 = c2 If we set c1 = c2 = c in (5), then the triangular decomposition method permits us to transform the equilibrium equations (5) into the following three triangular sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' T21 = [p1, p2], T22 = [p1 − 3 c, p2 − 3 c], T23 = [p2 1 − cp1 − c2, p2 + p1 − c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The zero of T21 is simply (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' From T23, we obtain two zeros1 ��√ 5 2 + 1 2 � c, � − √ 5 2 + 1 2 � c � , �� − √ 5 2 + 1 2 � c, �√ 5 2 + 1 2 � c � , which are useless as the component � − √ 5 2 + 1 2 � c is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Therefore, the only non-vanishing equilibrium is (3 c, 3 c), which can be obtained from T22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Let α = 1/2 and c1 = c2 = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The unique non-vanishing equilibrium (3 c, 3 c) is locally stable if c2 > 2 k1 + 2 k2 + � 4 k2 1 − 7 k1k2 + 4 k2 2 216 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The system may undergo a period-doubling bifurcation when c2 = 2 k1 + 2 k2 + � 4 k2 1 − 7 k1k2 + 4 k2 2 216 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Furthermore, there exist no other bifurcations of the equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Substituting p1 = 3 c and p2 = 3 c into J, we obtain that the Jacobian matrix at (3 c, 3 c) to be J(3 c, 3 c) = � 27 c2−k1 27 c2 k1 216 c2 k2 216 c2 27 c2−k2 27 c2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Consequently, Tr(J) = 54 c2 − k1 − k2 27 c2 , Det(J) = 5184 c4 − 192 c2k1 − 192 c2k2 + 7 k1k2 5184 c4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 1These zeros can also be obtained from T12 in (6) by setting c1 = c2 = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 5 One can verify that the first condition for the local stability is always fulfilled since k1, k2, c > 0 and CDJ 1 ≡ 1 − Tr(J) + Det(J) = 5 k1k2 3888 c4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The second condition is CDJ 2 ≡ 1 + Tr(J) + Det(J) = 15552 c4 + (−288 k1 − 288 k2) c2 + 5 k1k2 3888 c4 > 0, which means that 15552 c4 + (−288 k1 − 288 k2) c2 + 5 k1k2 > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=', c2 > 2 k1 + 2 k2 + � 4 k2 1 − 7 k1k2 + 4 k2 2 216 or c2 < 2 k1 + 2 k2 − � 4 k2 1 − 7 k1k2 + 4 k2 2 216 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The third condition is CDJ 3 ≡ 1 − Det(J) = (144 k1 + 144 k2) c2 − 5 k1k2 3888 c4 > 0, which implies that (144 k1 + 144 k2) c2 − 5 k1k2 > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=', c2 > 5 k1k2 144 (k1 + k2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Furthermore, it can be proved that 2 k1 + 2 k2 − � 4 k2 1 − 7 k1k2 + 4 k2 2 216 < 5 k1k2 144 (k1 + k2) < 2 k1 + 2 k2 + � 4 k2 1 − 7 k1k2 + 4 k2 2 216 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Accordingly, the equilibrium is locally stable if c2 > 2 k1 + 2 k2 + � 4 k2 1 − 7 k1k2 + 4 k2 2 216 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The rest of the proof follows immediately from Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Figure 1 depicts two 2-dimensional cross-sections of the stability region reported in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' It is observed that an increase in the marginal cost c or a decrease in the adjustment speeds k1, k2 has an effect of stabilizing the unique non-vanishing equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' (a) k2 = 1/10 (b) c = 1/3 Figure 1: The 2-dimensional cross-sections of the stability region of the considered model with α = 1/2 and c1 = c2 = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The curves of CDJ 2 = 0 and CDJ 3 = 0 are marked in blue and green, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 The general case If c1 ̸= c2, then the analytical expression of the unique non-vanishing equilibrium would be quite complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Thus, the proof of Theorem 1 can not work since it is impossible to substitute the analytical expression of the equilibrium into the Jacobian matrix and obtain a neat result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Concerning the bifurcation analysis, we need to determine the conditions on the parameters that CDJ 1 = 0, CDJ 2 = 0, and CDJ 3 = 0 are satisfied at the non-vanishing equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' For this purpose, the following notation is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Let A = m � i=0 ai xi, B = l � j=0 bj xj be two univariate polynomials in x with coefficients ai, bj, and am, bl ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The determinant ��������������� am am−1 · · a0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' am am−1 · · a0 bl bl−1 · · b0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' bl bl−1 · · b0 ��������������� � � � l � � � m is called the Sylvester resultant (or simply resultant) of A and B with respect to x, and denoted by res(A, B, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The following lemma reveals the main property of the resultant, which can also be found in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Let A and B be two univariate polynomials in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' There exist two polynomials F and G in x such that FA + GB = res(A, B, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Furthermore, A and B have common zeros in the field of complex numbers if and only if res(A, B) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' For a triangular set T = [T1(x), T2(x, y)] and a polynomial H(x, y), we define res(H, T ) ≡ res(res(H, T2, y), T1(x), x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' By Lemma 1, if T1 = 0 and T2 = 0 (or simply denoted as T = 0), then one knows that H = 0 implies res(H, T ) = 0, which means res(H, T ) = 0 is a necessary condition for H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Consequently, the following proposition is acquired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' It should be emphasized that Proposition 2 only reports the results for the case of k1 = k2 because the conditions for k1 ̸= k2 are too long to list in this paper due to space limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' However, readers can see that the idea of the proof also works for k1 ̸= k2 and can derive the complete conditions themself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Let α = 1/2 and k1 = k2 = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The system may undergo a period-doubling bifurcation when R1 = 0 and a Neimark-Sacker bifurcation when R2 = 0, where R1 and R2 are given in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' It should be noted that the resultant is feasible only for polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' For CDJ 1 , we consider its numerator Num(CDJ 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Then one can obtain that res(Num(CDJ 1 ), T12) = 81 k6c18 1 c6 2 (c1 + c2) � 32c2 1 + 61c1c2 + 32c2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Since c1 > 0, c2 > 0, and k > 0, it is impossible that res(Num(CDJ 1 ), T12) = 0 or CDJ 1 = 0 provided that T12 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Hence, the equilibrium can not lose its stability through a fold bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Furthermore, we have res(Num(CDJ 2 ), T12) = −729 c32 1 c8 2(c1 + c2)R1, res(Num(CDJ 3 ), T12) = 729 k3c32 1 c8 2(c1 + c2)R2, which will vanish only if R1 = 0 and R2 = 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Consequently, the system may undergo a period-doubling bifurcation when R1 = 0 and a Neimark-Sacker bifurcation when R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 7 By Proposition 1, there exists only one equilibrium (p1, p2) with p1 > 0 and p2 > 0 although its analytical expression is complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' To explore the local stability, we need to determine the signs of CDJ 1 , CDJ 2 , and CDJ 3 at this equilibrium without using its closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' It should be noted that CDJ 1 , CDJ 2 , and CDJ 3 are rational functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Suppose that CDJ i = Num(CDJ i ) Den(CDJ i ) , where Num(·) and Den(·) denote the numerator and the denominator, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Then the sign of CDJ i is the same as that of Num(CDJ i ) · Den(CDJ i ) if Den(CDJ i ) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' One could compute that res(Num(CDJ 1 ) · Den(CDJ 1 ), T12) = −1594323 k6c50 1 c17 2 (c1 + c2)6(32 c2 1 + 61 c1c2 + 32 c2 2), res(Num(CDJ 2 ) · Den(CDJ 2 ), T12) = 129140163 c70 1 c22 2 (c1 + c2)6R1, and res(Num(CDJ 3 ) · Den(CDJ 3 ), T12) = −129140163 k3c70 1 c22 2 (c1 + c2)6R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' We should emphasize that the sign of res(Num(CDJ i )·Den(CDJ i ), T12) may not be the same as Num(CDJ i )· Den(CDJ i ) or CDJ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' However, it is known that res(Num(CDJ i )·Den(CDJ i ), T12) involves only the parameters and its zeros divide the parameter space into several regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' In each region, the sign of CDJ i is invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Consequently, we just need to select one sample point from each region and identify the sign of CDJ i at the selected sample point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The selection of sample points might be extremely complicated in general and could be automated using, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=', the PCAD method [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' In Table 1, we list all the selected sample points and the corresponding information on whether the non-vanishing equilibrium is stable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=', whether CDJ 1 > 0, CDJ 2 > 0, and CDJ 3 > 0 are simultaneously satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Moreover, Table 1 displays the signs of R1 and R2 at these sample points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' One can observe that the equilibrium is stable if R1 > 0 and R2 > 0, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' It should be mentioned that the calculations involved in Table 1 are exact and rigorous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' That is, the computational results provide theoretical foundations for a systematic analysis of the local stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Therefore, we acquire the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' If k1 = k2 = k, the unique non-vanishing equilibrium (p1, p2) with p1 > 0 and p2 > 0 is locally stable if R1 > 0 and R2 > 0, where R1 and R2 can be found in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Table 1: Selected Sample Points in {(c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' k) | c1 > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' c2 > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' k > 0} for α = 1/2 (c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' k) stable R1 R2 (c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' k) stable R1 R2 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 1/4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 1) yes + + (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 5/16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 1) yes + + (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 1/4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 7) no − + (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 5/16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 10) no − + (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 1/4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 29) no − − (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 5/16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 30) no − − (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 1/4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 51) no + − (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 5/16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 51) no + − (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 1) yes + + ([1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 7/8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 1) yes + + (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 18) no − + (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 7/8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 38) no − + (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 35) no − − (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 7/8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 51) no − − (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 53) no + − (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 7/8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 65) no + − (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 9/8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 1) yes + + (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} 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+page_content=' 66) no − − (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 140) no − − (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 9/8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 83) no + − (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 209) no + − (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 1) yes + + (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 1) yes + + (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 91) no − + (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 112) no − + (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 272) no − − (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 462) no − − (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 453) no + − (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 811) no + − 8 4 α = 1/3 If α = 1/3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' then β = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' We have the iteration map � � � � � � � � � � � � � � � � � � � p1(t + 1) = p1(t) + k1 −p1(t) � p1(t)p2(t) + � 2 p2(t) + 3 � p1(t)p2(t) � c1 2 p2 1(t) �� p1(t) + � p2(t) �2 , p2(t + 1) = p2(t) + k2 −p2(t) � p1(t)p2(t) + � 2 p1(t) + 3 � p1(t)p2(t) � c2 2 p2 2(t) �� p1(t) + � p2(t) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' (7) By setting p1(t + 1) = p1(t) = p1 and p2(t + 1) = p2(t) = p2, one can obtain the equations of the equilibrium � − p1 √p1p2 + (2 p2 + 3√p1p2) c1 = 0, − p2 √p1p2 + (2 p1 + 3√p1p2) c2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Denote √p1 = x and √p2 = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The above equations become � − x3y + (2 y2 + 3 xy)c1 = 0, − y3x + (2 x2 + 3 xy)c2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' (8) Using the triangular decomposition method, we decompose the solutions of system (8) into zeros of the following two triangular sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' T31 = [x, y] , T32 = � x8 − 9 c1x6 + 27 c2 1x4 + (−27 c3 1 − 12 c2 1c2)x2 + 20 c3 1c2, 2 c1y − x3 + 3 c1x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Evidently, T31 is corresponding to the origin (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Therefore, the identification of the number of non- vanishing equilibria can be transformed into the determination of the number of real solutions of the following semi-algebraic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' � � � � � x8 − 9 c1x6 + 27 c2 1x4 + (−27 c3 1 − 12 c2 1c2)x2 + 20 c3 1c2 = 0, 2 c1y − x3 + 3 c1x = 0, x > 0, y > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Using the algebraic approach by Li and Wang [25], we know that the above system has one unique real solution for any parameter values of c1, c2 > 0, which implies the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Let α = 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The iteration map (7) possesses one unique equilibrium (p1, p2) with p1 > 0 and p2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' To investigate the local stability of the equilibrium,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' we consider the Jacobian matrix M = �M11 M12 M21 M22 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' where M11 = 12 p 9 2 1 √p2 + 4 p 7 2 1 p 3 2 2 − 15 c1k1p 3 2 1 √p2 − 8 c1k1p 3 2 2 √p1 + 3 k1p 5 2 1 √p2 + 4 p5 1 + 12 p4 1p2 − 21 c1k1p1p2 + k1p2 1p2 4 p 7 2 1 �√p1 + √p2 �3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' M12 = k1 �√p2 p 3 2 1 − p2 1 + c1√p2√p1 + 3 p1c1 � 4 p2 1 �√p1 + √p2 �3 √p2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' M21 = k2 �√p1 p 3 2 2 + c2√p2√p1 + 3 p2c2 − p2 2 � 4 p2 2 �√p1 + √p2 �3 √p1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 9 M22 = 4 p 3 2 1 p 7 2 2 + 12 p 9 2 2 √p1 − 8 c2k2p 3 2 1 √p2 − 15 c2k2p 3 2 2 √p1 + 3 k2p 5 2 2 √p1 + 12 p1 p4 2 + 4 p5 2 − 21 c2k2p1p2 + k2p1p2 2 4p 7 2 2 �√p1 + √p2 �3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' As in Section 3, we denote CDM 1 ≡ 1 − Tr(M) + Det(M), CDM 2 ≡ 1 + Tr(M) + Det(M), CDM 3 ≡ 1 − Det(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1 The special case of c1 = c2 If we set c1 = c2 = c, then the triangular decomposition method permits us to transform the equilibrium equations (8) into the following triangular sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' T41 = [x, y], T42 = [x2 − c, y + x], T43 = [x2 − 5 c, y − x], T44 = [x4 − 3 c x2 + 4 c2, 2 cy − x3 + 3 cx].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Obviously, the zeros of T41 and T42 are economically uninteresting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Moreover, all the roots of x4−3 c x2+ 4 c2 of T44, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=', � 2 √ 7c i + 6 c 2 , − � 2 √ 7c i + 6 c 2 , � −2 √ 7c i + 6 c 2 , − � −2 √ 7c i + 6 c 2 , are imaginary and also not of our concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' There exists only one non-vanishing equilibrium (p1, p2) = (5 c, 5 c), which is corresponding to the branch T43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Substituting p1 = 5 c and p2 = 5 c into M, we obtain the Jacobian matrix at the equilibrium (5 c, 5 c) to be M(5 c, 5 c) = � 500 c2−3 k1 500 c2 k1 1000 c2 k2 1000 c2 500 c2−3 k2 500 c2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Hence, Tr(M) = 1000 c2 − 3 k1 − 3k2 500 c2 , Det(M) = 200000 c4 − 1200 c2k1 − 1200 c2k2 + 7 k1k2 200000 c4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Let α = 1/3 and c1 = c2 = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The unique non-vanishing equilibrium (5 c, 5 c) is locally stable if c2 > 3 k1 + 3 k2 + � 9 k2 1 − 17 k1k2 + 9 k2 2 2000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The system may undergo a period-doubling bifurcation when c2 = 3 k1 + 3 k2 + � 9 k2 1 − 17 k1k2 + 9 k2 2 2000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Furthermore, there exist no other bifurcations of the equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The first condition for the local stability is always fulfilled since CDM 1 ≡ 1 − Tr(M) + Det(M) = 7 k1k2 200000c4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The second condition should be CDM 2 ≡ 1 + Tr(M) + Det(M) = 800000 c4 + (−2400 k1 − 2400 k2) c2 + 7 k1k2 200000 c4 > 0, 10 which implies that 800000 c4 + (−2400 k1 − 2400 k2) c2 + 7 k1k2 > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=', c2 > 3 k1 + 3 k2 + � 9 k2 1 − 17 k1k2 + 9 k2 2 2000 or c2 < 3 k1 + 3 k2 − � 9 k2 1 − 17 k1k2 + 9 k2 2 2000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The third condition should be CDM 3 ≡ 1 − Det(M) = (1200 k1 + 1200 k2) c2 − 7 k1k2 200000 c4 > 0, from which we have (1200 k1 + 1200 k2) c2 − 7 k1k2 > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=', c2 > 7 k1k2 1200 (k1 + k2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' It can be proved that 3 k1 + 3 k2 − � 9 k2 1 − 17 k1k2 + 9 k2 2 2000 < 7 k1k2 1200 (k1 + k2) < 3 k1 + 3 k2 + � 9 k2 1 − 17 k1k2 + 9 k2 2 2000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Therefore, the equilibrium is locally stable if c2 > 3 k1 + 3 k2 + � 9 k2 1 − 17 k1k2 + 9 k2 2 2000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The rest of the proof follows from Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' In Figure 2, we show two 2-dimensional cross-sections of the stability region reported in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' One can see that the equilibrium may lose its stability if the adjustment speeds k1, k2 are large enough or the marginal cost c is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' (a) k2 = 1/10 (b) c = 1/10 Figure 2: The 2-dimensional cross-sections of the stability region of the considered model with α = 1/3 and c1 = c2 = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The curves of CDM 2 = 0 and CDM 3 = 0 are marked in blue and green, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 The general case As in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2, we set k1 = k2 = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' We should mention that the method employed in this section also works for the case of k1 ̸= k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' However, the conditions for k1 ̸= k2 are tedious and not reported in this section due to space limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Interested readers can use our method to compute the complete conditions themself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The case of c1 = c2 has been explored in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1, hence we suppose that c1 ̸= c2 in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The bifurcations are analyzed in the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 11 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Let α = 1/3, k1 = k2 = k and c1 ̸= c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The iteration map (7) may undergo a period- doubling bifurcation when R3 = 0 and a Neimark-Sacker bifurcation when R4 = 0, where R3 and R4 are given in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Computing the resultant of Num(CDM 1 ) with respect to T32, one obtains res(Num(CDM 1 ), T32) = 879609302220800000 k16c51 1 c11 2 (c1 − c2)2 � 2187 c2 1 − 4031 c1c2 + 2187 c2 2 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' It is evident that 2187 c2 1 − 4031 c1c2 + 2187 c2 2 = 2187(c1 − c2)2 + 343 c1c2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Therefore, res(Num(CDM 1 ), T32) ̸= 0, which means that CDM 1 ̸= 0 at the unique non-vanishing equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Hence, there exist no fold bifurcations in map (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Furthermore, we have res(Num(CDJ 2 ), T32) = 99035203142830421991929937920000000 c101 1 c13 2 (c1 − c2)2 R2 3, and res(Num(CDJ 3 ), T32) = 99035203142830421991929937920000000 k8c101 1 c13 2 (c1 − c2)10R2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Consequently, a period-doubling bifurcation may occur when R3 = 0, while a Neimark-Sacker bifurcation may take place when R4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' To investigate the local stability, we need to consider Num(CDJ i ) · Den(CDJ i ) and compute its resultant with respect to T32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Then it is obtained that res(Num(CDJ 1 ) · Den(CDJ 1 ), T32) = 5708990770823839524233143877797980545530986496 · 1020 k16c156 1 c36 2 (c1 − c2)12(2187 c2 1 − 4031 c1c2 + 2187 c2 2)2, res((Num(CDJ 2 ) · Den(CDJ 2 ), T32) = 6582018229284824168619876730229402019930943462534319453394436096 · 1024 c218 1 c42 2 (c1 − c2)10R2 3, res((Num(CDJ 3 ) · Den(CDJ 3 ), T32) = 6582018229284824168619876730229402019930943462534319453394436096 · 1024 k8c218 1 c42 2 (c1 − c2)10R2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' These res(Num(CDJ i )·Den(CDJ i ), T32) involve only the parameters and their zeros divide the parameter set {(c1, c2, k) | c1 > 0, c2 > 0, k > 0} into several regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' In each region, the signs of CDM 1 , CDM 2 , and CDM 3 are fixed and can be identified by checking at a selected sample point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' In Table 2, we list the 40 selected sample points and the signs of R3, R4 at these sample points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Moreover, Table 2 provides the information on whether the non-vanishing equilibrium is stable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=', whether the stability conditions CDM 1 > 0, CDM 2 > 0 and CDM 3 > 0 are satisfied simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Interested readers may check the correctness of Table 2 themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Based on a series of computations, we acquire the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Let k1 = k2 = k and c1 ̸= c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The unique non-vanishing equilibrium of map (7) is locally stable if one of the following conditions is satisfied: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' R3 > 0, R4 > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' R3 < 0, R4 > 0 and A1 > 0, A2 < 0, A3 > 0, where R3, R4, A1, A2, and A3 can be found in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' From Table 2, we see that the equilibrium is stable if R3 > 0 and R4 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Hence, R3 > 0, R4 > 0 is a sufficient condition for the local stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' However, this condition is not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' For example, at the first sample point (1, 1/4, 1/512) listed in Table 2, the equilibrium is locally stable, but one can verify that R3 < 0 and R4 > 0 at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Thus, the second condition of Theorem 4 is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The necessity of the second condition can also be illustrated by Figure 4 (b, d, f), where the regions defined by the first and second conditions are marked in light grey and dark grey, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' By economic 12 intuition, we know that for a fixed value of the marginal cost c2, a decrease in the adjustment speed k would be beneficial to the local stability of the equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' That is to say, the dark grey regions defined by the second condition would be more likely to be included in the stability regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' It is noted that A1, A2, and A3 involved in the second condition are contained in the so-called generalized discriminant list and can be picked out by repeated trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Concerning the generalized discriminant list, the reader may refer to [38] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The polynomials A1, A2, and A3 are needed here since the condition that R3 < 0 and R4 > 0 is not a sufficient condition for the local stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' For example, the model is stable at (1, 1/4, 1/512), where R3 < 0 and R4 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' But, the model is unstable at (1, 1/4, 34), where R3 < 0 and R4 > 0 are also satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Consequently, additional polynomials are needed to constrict the region defined by R3 < 0 and R4 > 0 such that the complete stability conditions can be acquired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Table 2: Selected Sample Points in {(c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' k) | c1 > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' c2 > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' k > 0} for α = 1/3 (c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' k) stable R3 R4 (c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' k) stable R3 R4 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 1/4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 1/512) yes − + (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 3/8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 1/128) yes − + (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 1/4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 1) yes + + (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 3/8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 1) yes + + (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 1/4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 34) no − + (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 2536) no + − 5 Influence of the Substitutability Degree Firstly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' we analyze the influence of the substitutability degree α on the size of the stability region of the equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' We start by considering the special case of c1 = c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Let c1 = c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The stability region for α = 1/2 is a proper subset of that for α = 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Recall Theorems 1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' We need to prove that 2 k1 + 2 k2 + � 4 k2 1 − 7 k1k2 + 4 k2 2 216 > 3 k1 + 3 k2 + � 9 k2 1 − 17 k1k2 + 9 k2 2 2000 , which is equivalent to � 2 k1 + 2 k2 + � 4 k2 1 − 7 k1k2 + 4 k2 2 216 �2 − � 3 k1 + 3 k2 + � 9 k2 1 − 17 k1k2 + 9 k2 2 2000 �2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The left-hand side of the above inequality can be simplified into − (4374 k1 + 4374 k2) � 9 k2 1 − 17 k1k2 + 9 k2 2 2916000000 + (250000 k1 + 250000 k2) � 4 k2 1 − 7 k1k2 + 4 k2 2 2916000000 13 + 243439 k2 1 1458000000 + 61771 k1k2 2916000000 + 243439 k2 2 1458000000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' It is easy to check that (4374 k1 + 4374 k2) � 9 k2 1 − 17 k1k2 + 9 k2 2 2916000000 < (250000 k1 + 250000 k2) � 4 k2 1 − 7 k1k2 + 4 k2 2 2916000000 , which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' If c1 ̸= c2, however, the conclusion of the above proposition would be incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' For example, if we assume k1 = k2 = k and take (c1, c2, k) = (261/65536, 1/2, 79/1024), then R1 = 588713082686404258452596575293972215811486125608829 6129982163463555433433388108601236734474956488734408704 > 0, R2 = 108130364702270905134254005155560019343 340282366920938463463374607431768211456 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Hence, (261/65536, 1/2, 79/1024) is in the stability region of the model for α = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' But,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' at the same parameter point,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' namely (c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' k) = (261/65536,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 79/1024),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' we have R3 = − 791461358900213183480020700044263844445257635142615074110540187 26328072917139296674479506920917608079723773850137277813577744384 < 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' R4 = 526438846625624761986017962528229497389068363385599391 374144419156711147060143317175368453031918731001856 > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' and A1 = 44864955 4294967296 > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' A2 = − 842240947483983714275440267 81129638414606681695789005144064 < 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' A3 = − 63936547182666560163845458457577 649037107316853453566312041152512 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' This means that the stability conditions of Theorem 4 for α = 1/3 are not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' On the other hand, one can also find some points where the model is stable for α = 1/3 but unstable for α = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' For example, at (c1, c2, k) = (3/8, 1/2, 827/64), we know R3 = 40079185741889580295152003015 288230376151711744 > 0, R4 = 29339436396656781 17179869184 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Therefore, (3/8, 1/2, 827/64) is in the stability region for α = 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' However, R1 = −24200272602071108539 17592186044416 < 0, R2 = −96467864887 67108864 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' That is to say, (3/8, 1/2, 827/64) is an unstable parameter point for α = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Figure 3 depicts the 2-dimensional cross-sections of the stability regions for α = 1/2 and α = 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' For comparison purposes, we place the cross-sections for α = 1/2 on the left and those for α = 1/3 on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' We set k1 = k2 = k and choose three different values of the parameter k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=', k = 1/2, 1, 10, to observe the effect of variation of k on the size of the stability regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The curves of R1 = 0 and R3 = 0 are marked in blue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' the curves of R2 = 0 and R3 are marked in green;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' the curves of A1 = 0, A2 = 0 and A3 = 0 are marked in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The stability regions are colored in light grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' From Figure 3, we find that the stability region would shrink if the firms react or adjust their outputs faster both for α = 1/2 and α = 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Similarly, in Figure 4, we assume that k1 and k2 are identical and choose three different values of c1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=', c1 = 1/2, 1, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The regions of R1 > 0, R2 > 0 and those of R3 > 0, R4 > 0 are colored in light grey, while the regions defined by R3 < 0, R4 > 0, A1 > 0, A2 < 0, A3 > 0 are colored in dark grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' From Figure 4, we observe that increasing the marginal cost c1 of the first firm could result in the enlargement of the stability region for α = 1/2 and α = 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' As aforementioned, in the case of c1 ̸= c2 and k1 = k2, it can not be proved that the stability region for α = 1/3 covers that for α = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' From Figures 3 and 4, however, it seems that the stability region for α = 1/3 is larger than that for α = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Consequently, for the Bertrand duopoly model considered in this paper, we may conclude that increasing the substitutability degree α has an effect of destabilizing the unique 14 non-vanishing equilibrium in some sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' In other words, product differentiation might make the considered model more stable, which is an important finding from an economic point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Shy [34] discussed the traditional view on the degree of product differentiation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=', a decrease in product differentiation may result in an increase in market competition intensity and even a price war among involved firms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The possible explanation for our finding is that a price war might destabilize the equilibrium of the Bertrand game with differentiated goods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' It should be noted that our conclusion is in contrast with the one by Agliari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Specifically, Agliari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' [1] investigated a Cournot duopoly model with differentiated products and employed the same CES utility function and the same linear cost functions as in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' However, they discovered that a higher degree of product differentiation or a lower degree of substitutability leads to the destabilization of their model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' This contradiction may help reveal the essential difference between the Bertrand and Cournot oligopolies with differentiated goods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' From an economic point of view, the effects on economic variables such as prices and profits of changing the substitutability degree are interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' In the sequel, we focus on the comparative statics in the special case of identical marginal costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Let c1 = c2 = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' According to (3), the equilibrium satisfies that � − pβ 2p1+β 1 β + p2β 2 c + (pβ 1pβ 2)(1 + β)c = 0, − pβ 1p1+β 2 β + p2β 1 c + (pβ 1pβ 2)(1 + β)c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' (9) Hence, −pβ 2p1+β 1 β + p2β 2 c = −pβ 1p1+β 2 β + p2β 1 c, which implies that (p2β 1 − p2β 2 )c = (p2 − p1)pβ 1pβ 2β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Without loss of generality, we suppose that p1 ≥ p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Since c > 0 and β > 0, we know (p2β 1 − p2β 2 )c ≥ 0 and (p2 − p1)pβ 1pβ 2β ≤ 0, which implies p1 = p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Plugging p1 = p2 into the first equation of (9), one can solve p1 = p2 = c(2+β) β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Therefore, at the equilibrium q1 = q2 = β 2 c(2+β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' As β = α/(1 − α), we obtain ∂pi ∂α = −2 c α2 < 0, ∂qi ∂α = 1 (−2 + α)2 c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' According to (2), the profits of the two firms would be Π1 = Π2 = �c(2 + β) β − c � β 2 c(2 + β) = 1 2 + β = 1 + 1 α − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Hence, for i = 1, 2, ∂Πi ∂α = − 1 (α − 2)2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Recalling the inverse demands (1), for a point (q∗ 1, q∗ 2) on the indifference curve, we define the consumer surplus of the first product to be CS1 = � q∗ 1 0 qα−1 1 qα 1 + q∗α 2 dq1 = 1 α � q∗ 1 0 d(qα 1 + q∗α 2 ) qα 1 + q∗α 2 = 1 α ln � 1 + �q∗ 1 q∗ 2 �α� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' In the case of c1 = c2, the outputs of the two products are equal at the equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Therefore, we have that CS1 = CS2 = 1 α ln 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Accordingly, the social welfare is W = CS1 + CS2 + Π1 + Π2 = 2 α ln 2 + 2 α − 2 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Then it is known that ∂W ∂α = −2 ln 2 α2 − 2 (α − 2)α < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' To summarize, in the special case of identical marginal costs, an increase in the substitutability degree α leads to a stable equilibrium with lower prices, higher supplies, lower profits, and lower welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' In other words, the degree of product differentiation is positively related to the prices of the goods, the profits of the involved companies, and the social welfare, which is consistent with our economic intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 15 (a) α = 1/2, k = 1/2 (b) α = 1/3, k = 1/2 (c) α = 1/2, k = 1 (d) α = 1/3, k = 1 (e) α = 1/2, k = 10 (f) α = 1/3, k = 10 Figure 3: The 2-dimensional cross-sections of the stability regions for α = 1/2 and α = 1/3 if we set k1 = k2 = k and fix k = 1/2, 1, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The curves of R1 = 0 and R3 = 0 are marked in blue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' the curves of R2 = 0 and R3 are marked in green;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' the curves of A1 = 0, A2 = 0 and A3 = 0 are marked in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The stability regions are colored in light grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 16 (a) α = 1/2, c1 = 1/2 (b) α = 1/3, c1 = 1/2 (c) α = 1/2, c1 = 1 (d) α = 1/3, c1 = 1 (e) α = 1/2, c1 = 10 (f) α = 1/3, c1 = 10 Figure 4: The 2-dimensional cross-sections of the stability regions for α = 1/2 and α = 1/3 if we set k1 = k2 = k and fix c1 = 1/2, 1, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The curves of R1 = 0 and R3 = 0 are marked in blue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' the curves of R2 = 0 and R3 are marked in green;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' the curves of A1 = 0, A2 = 0 and A3 = 0 are marked in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The regions of R1 > 0, R2 > 0 and those of R3 > 0, R4 > 0 are colored in light grey, while the regions defined by R3 < 0, R4 > 0, A1 > 0, A2 < 0, A3 > 0 are colored in dark grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 17 6 Numerical Simulations This section provides numerical simulations to illustrate the complex dynamics of the considered Bertrand duopoly model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The first purpose of our simulations is to confirm the main conclusion of Section 5 that increasing the substitutability degree α could destabilize the unique non-vanishing equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' In Figure 5, we depict the 1-dimensional bifurcation diagrams with respect to α, where we fix the other parameters k1 = k2 = 1, c1 = c2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 and set the initial point to be (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='56, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='06).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The bifurcation diagrams against p1 and p2 are given in Figure 5 (a, c) and (b, d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' It is observed that complex dynamics appear when α becomes large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Specifically, there exists one unique stable equilibrium at first, then a stable 2-cycle orbit, and finally a chaotic set as α varies from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1 up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' To show the transition clearly, the 1-dimensional bifurcation diagrams are enlarged for α ∈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='55, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='6) in (c, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' One can see that, when α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='553372, a branching point occurs and the unique fixed point bifurcates into a 2-cycle orbit, which, however, is not a period-doubling bifurcation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' This 2-cycle orbit loses its stability through a Neimark- Sacker bifurcation rather than a period-doubling bifurcation at α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='577570.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' More details can be found in Figure 6, where we plot the phase portraits for k1 = k2 = 1 and c1 = c2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 with the initial point (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='56, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='06).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' From Figure 6 (a), we observe that, after the occurrence of a Neimark- Sacker bifurcation, the 2-cycle orbit (P21(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='464194, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='607384) and P21(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='607384, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='464194)) becomes unstable and bifurcates into two invariant closed orbits when α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='58;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' the unique equilibrium E1(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='492557, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='492557) goes to E1new(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='489655, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='489655) when α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Furthermore, all points on the diagonal line x = y converge to E1new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The two invariant closed orbits marked in blue are stable and points converge to them from inside and outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Figure 6 (b) depicts the phase portrait when α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='59 and the other parameters are set to be the same as (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' From (b), one can discover chaotic attractors with symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The above observations show that an increase in the substitutability degree α leads to the emergence of instability, complex dynamics, and even chaos in the considered model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' (a) against p1 (b) against p2 (c) against p1 and enlarged for α ∈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='55, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='6) (d) against p2 and enlarged for α ∈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='55, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='6) Figure 5: The 1-dimensional bifurcation diagrams with respect to α if we fix k1 = k2 = 1, c1 = c2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 and set the initial point to be (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='56, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='06).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 18 4 pi 3 2 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='4 a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='77 9 54 3 2 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='4 L a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='77 9 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='7 pi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='6 NS BP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='5 NS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='58 a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='6 NS BP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='5 NS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='58 a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='7 p1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='7 p2 (a) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='8 p1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='8 p2 (b) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='59 Figure 6: Phase portraits for k1 = k2 = 1 and c1 = c2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 with the initial point (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='56, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='06).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' To illustrate the influence of other parameters, several 2-dimensional bifurcation diagrams are computed and displayed in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Figure 7 depicts the 2-dimensional bifurcation diagram of map (4) (α = 1/2) with respect to k1 and k2 if we fix c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='3, c2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='4 and set the initial point to be (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' We detect periodic orbits with distinct orders and mark the corresponding parameter points in different colors in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' It should be mentioned that the parameter points where there exist periodic orbits with orders more than 25 are marked in light yellow as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Two different routes from the unique stable equilibrium to complex dynamics can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' For example, if we fix k2 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='5 and change the value of k1 from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='0 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='0, the dynamics of the system start from one unique stable equilibrium (the dark blue region), then transition to a stable 2-cycle orbit (the light blue region) and finally to invariant closed orbits as well as chaos (the light yellow region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' This is similar to the route displayed in Figure 5, where the stable 2-cycle loses its stability through a Neimark-Sacker bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The other route can be discovered, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=', if we fix k2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='5 and keep k1 as a free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Then it is observed that the unique stable equilibrium loses its stability through a cascade of period-doubling bifurcations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' In Figure 8, we plot the 2-dimensional bifurcation diagram of map (7) (α = 1/3) with respect to k1 and k2 if fixing c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1, c2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='15 and setting the initial point to be (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Similar to Figure 7, the aforementioned two routes from local stability to complex dynamics can also be observed in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The 2-dimensional bifurcation diagrams with respect to c1 and c2 for α = 1/2 and α = 1/3 are displayed in Figures 9 and 10, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' One can see that complicated dynamic phenomena take place if one of the cost parameters c1, c2 is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Similarly, we find the above two routes to chaotic behavior, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=', through a cascade of period-doubling bifurcation and through a Neimark-Sacker bifurcation on a 2-cycle orbit, which have already been discovered by Ahmed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' However, from Figure 9, we also find the existence of a Neimark-Sacker bifurcation directly on the unique equilibrium, which is a new result that has not been observed by Ahmed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' [3] yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Specifically, Figure 9 shows that, if we fix c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='9 and decrease the value of c2 from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='0, the dynamics of the system directly transition from the unique stable equilibrium (the dark blue region) to invariant closed orbits (the light yellow region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' In this case, the behavior of the market suddenly changes from an ordered state to a disordered state at some critical point, which can hardly be learned by even rational players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 7 Concluding Remarks In this paper, we investigated the local stability, bifurcations, and comparative statics of a dynamic Bertrand duopoly game with differentiated products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' This duopoly is assumed to possess two boundedly rational players adopting a gradient adjustment mechanism and a continuum of identical consumers with a CES utility function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Moreover, the cost functions are supposed to be linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' It should be mentioned that the nonlinearity of the resulting demand function derived from the underlying utility permits us to extend the applications of Bertrand games to more realistic economies, compared to the widely used Bertrand models with linear demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The considered game was first explored by Ahmed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' [3], where only numerical simulations are 19 Figure 7: The 2-dimensional bifurcation diagram of map (4) (α = 1/2) with respect to k1 and k2 if we fix c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='3, c2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='4 and set the initial point to be (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Figure 8: The 2-dimensional bifurcation diagram of map (7) (α = 1/3) with respect to k1 and k2 if we fix c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1, c2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='15 and set the initial point to be (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 20 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='00 25 24 23 22 21 20 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='50 19 18 17 16 15 14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='00 13 12 11 10 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='50 6 5 3 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='50 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='00 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='50 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='00 k16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='00 25 24 23 22 21 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='50 19 18 17 16 15 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='00 13 12 11 10 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='50 6 5 3 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='50 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='00 k1Figure 9: The 2-dimensional bifurcation diagram of map (4) (α = 1/2) with respect to c1 and c2 if we fix k1 = 6, k2 = 12 and set the initial point to be (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Figure 10: The 2-dimensional bifurcation diagram of map (7) (α = 1/3) with respect to c1 and c2 if we fix k1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='3, k2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='6 and set initial point to be (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 21 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 25 24 23 22 21 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 19 18 17 16 15 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='50 13 12 11 10 9 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 6 5 4 3 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='00 C10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='10 24 25 24 23 22 21 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='08 19 18 17 16 15 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='05 13 12 11 10 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='03 5 4 3 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='10 C1employed to investigate the dynamic behavior and it was observed that the Nash equilibrium loses its stability through a period-doubling bifurcation as the speed of adjustment increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' In our study, however, we re-investigated this game using several tools based on symbolic computations such as the triangular decomposition method (refer to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=', [23]) and the PCAD method (refer to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=', [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The results of symbolic computations are exact, and thus provide theoretical foundations for the systematic analysis of economic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' For simplicity, our work mainly focused on two specific degrees of product substitutability, namely α = 1/2 and α = 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' In both cases, we proved the uniqueness of the non-vanishing equilibrium using the algebraic approach of detecting the multiplicity of equilibria proposed by the first author and his co-worker [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' We introduce several tools based on symbolic computations and used them to obtain the rigorous conditions for the local stability of the unique non-vanishing equilibrium for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' In the special case that the two firms have identical marginal costs, we proved that the model can lose its stability only through a period-doubling bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' From an economic point of view, the most interesting finding was that an increase in the substitutability degree or a decrease in the product differentiation leads to the destabilization of the Bertrand model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' This is because a price war, which might destabilize the equilibrium, can take place if the substitutability degree is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' We should mention that our finding is in contrast with that by Agliari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' [1] and that by Fanti and Gori [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' This contradiction contributes to the literature on the connection between Cournot and Bertrand oligopolies and may help reveal the essential difference between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Moreover, we conducted the comparative statics in the special case of identical marginal costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The resulting conclusion was that lower degrees of product differentiation mean lower prices, higher supplies, lower profits, and lower social welfare, which is consistent with our economic intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Numerical simulations were provided in the end, through which complex dynamics such as periodic orbits and chaos can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The simulations confirmed that an increase in the substitutability degree α leads to the emergence of instability, complex dynamics, and even chaos in the considered model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Two- dimensional bifurcation diagrams were also provided to show different possible routes to chaotic behavior, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=', through a cascade of period-doubling bifurcation and through a Neimark-Sacker bifurcation on a 2-cycle orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Furthermore, we discovered the existence of a Neimark-Sacker bifurcation directly on the equilibrium, which is a new finding and has not yet been discovered by Ahmed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='Appendix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='R1 = 15552 c10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1 c6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 + 62208 c9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 + 93312 c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 + 62208 c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 + 15552 c6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 + 73728 c11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1 c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k + 327168 c10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1 c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='+ 576576 c9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k + 541440 c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k + 436608 c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k + 541440 c6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k + 576576 c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k + 327168 c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='+ 73728 c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 k + 32768 c11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1 c2 k2 + 94208 c10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1 c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k2 + 284160 c9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k2 + 1163712 c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k2 + 2855520 c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='+ 3825168 c6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k2 + 2855520 c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k2 + 1163712 c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k2 + 284160 c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k2 + 94208 c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 k2 + 32768 c1c11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='+ 77824 c9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c2 k3 + 359936 c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k3 + 644608 c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k3 + 610976 c6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k3 + 494368 c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k3 + 610976 c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='+ 644608 c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k3 + 359936 c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k3 + 77824 c1c9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k3 − 4096c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1k4 − 12288 c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c2 k4 + 4544 c6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='+ 70360 c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k4 + 114600 c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k4 + 70360 c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k4 + 4544 c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k4 − 12288 c1c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k4 − 4096 c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='− 1024 c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c2 k5 − 3232 c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k5 − 4488 c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k5 − 3232 c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k5 − 1024 c1c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k5 + 32 c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c2 k6 + 61 c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='+ 32 c1c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' R2 = 1152 c8 1c2 2 + 5832 c7 1c3 2 + 12960 c6 1c4 2 + 16560 c5 1c5 2 + 12960 c4 1c6 2 + 5832 c3 1c7 2 + 1152 c2 1c8 2 + 1024 c8 1k + 3584 c7 1c2 k + 5920 c6 1c2 2k + 6224 c5 1c3 2k + 5836 c4 1c4 2k + 6224 c3 1c5 2k + 5920 c2 1c6 2k + 3584 c1c7 2k + 1024 c8 2k + 512 c5 1c2 k2 + 1616 c4 1c2 2k2 + 2244 c3 1c3 2k2 + 1616 c2 1c4 2k2 + 512 c1c5 2k2 − 32 c3 1c2 k3 − 61 c2 1c2 2k3 − 32 c1c3 2k3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='R3 = − 209715200000 c12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1 c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 + 838860800000 c11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1 c9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 − 1258291200000 c10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1 c10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 + 838860800000 c9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='− 209715200000 c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 + 1160950579200 c13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1 c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k − 5170397184000 c12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1 c6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k + 9284105011200 c11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1 c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='− 9178054656000 c10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1 c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k + 7806792499200 c9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k − 9178054656000 c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 k + 9284105011200 c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='− 5170397184000 c6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 k + 1160950579200 c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 k + 626913312768 c13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1 c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k2 − 1827529703424 c12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1 c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='+ 6377496477696 c11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1 c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k2 − 24562717922304 c10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1 c6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k2 + 56911413825536 c9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='− 74841436780544 c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k2 + 56911413825536 c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k2 − 24562717922304 c6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='+ 6377496477696 c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 k2 − 1827529703424 c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 k2 + 626913312768 c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 k2 − 117546246144 c12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1 c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='+ 2268751389696 c11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1 c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k3 − 8446241806848 c10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1 c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k3 + 13848228389376 c9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k3 − 12871123435008 c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='+ 10570707526656 c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k3 − 12871123435008 c6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k3 + 13848228389376 c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='− 8446241806848 c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 k3 + 2268751389696 c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 k3 − 117546246144 c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 k3 + 7346640384 c11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1 c2 k4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='+ 23872802112 c10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1 c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k4 − 79144786368 c9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k4 − 389232360000 c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k4 + 1762366805056 c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='− 2639431381760 c6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k4 + 1762366805056 c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k4 − 389232360000 c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k4 − 79144786368 c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='+ 23872802112 c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 k4 + 7346640384 c1c11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 k4 − 153055008 c10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1 k5 + 444048480 c9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c2 k5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='− 2281361760 c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k5 − 6359031360 c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k5 + 33853070112 c6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k5 − 51945109632 c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='+ 33853070112 c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k5 − 6359031360 c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k5 − 2281361760 c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k5 + 444048480 c1c9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='− 153055008 c10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 k5 + 36636624 c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c2 k6 − 65578896 c6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k6 + 239834412 c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k6 − 377249916 c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='+ 239834412 c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k6 − 65578896 c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k6 + 36636624c1c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k6 − 669222 c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c2 k7 + 1023534 c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='− 951468 c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k7 + 1023534 c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k7 − 669222 c1c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k7 + 2187 c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c2 k8 − 4031 c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k8 + 2187 c1c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='R4 = 17714700 c10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1 c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 − 84798900 c9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 + 166819500 c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 − 187523100 c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 + 175575600 c6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 − 187523100 c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='+ 166819500 c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 − 84798900 c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 + 17714700 c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 + 19131876 c10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1 k − 55506060 c9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c2 k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='+ 70441812 c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k − 70683840 c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k + 106503012 c6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k − 136123200 c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k + 106503012 c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='− 70683840 c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k + 70441812 c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k − 55506060 c1c9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k + 19131876 c10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2 k − 9159156 c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c2 k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='+ 23480604 c6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k2 − 24625107 c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k2 + 19286271 c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k2 − 24625107 c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k2 + 23480604 c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='− 9159156 c1c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k2 + 334611 c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c2 k3 − 511767 c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k3 + 475734 c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k3 − 511767 c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k3 + 334611 c1c5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='− 2187 c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c2 k4 + 4031 c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='1c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k4 − 2187 c1c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content='2k4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' A1 = 243 c2 1 + 352 c1c2 − 9 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' A2 = − 8000 c5 1c3 2 + 19683 c6 1k − 17496 c5 1c2 k + 3024 c4 1c2 2k + 1728 c3 1c3 2k − 2187 c4 1k2 + 3564 c3 1c2 k2 − 432 c2 1c2 2k2 + 81 c2 1k3 + 36 c1c2 k3 − k4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' A3 = 12754584 c7 1 − 12171384 c6 1c2 + 3708504 c5 1c2 2 + 84096 c4 1c3 2 + 2519424 c3 1c4 2 − 72171 c5 1k − 3576744 c4 1c2 k + 5126856 c3 1c2 2k − 629856 c2 1c3 2k − 25272 c3 1k2 + 98966 c2 1c2 k2 + 52488 c1c2 2k2 + 387 c1 k3 − 1458 c2 k3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Acknowledgements The authors wish to thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' Li Su for the beneficial discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' The authors are grateful to the anonymous referees for their helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' This work has been supported by Philosophy and Social Science Foundation of Guangdong (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' GD21CLJ01), Natural Science Foundation of Anhui Province (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} +page_content=' 2008085QA09), University Natural Science Research Project of Anhui Province (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAzT4oBgHgl3EQfFPpp/content/2301.01007v1.pdf'} 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+Environments +Zhongguo Li, Member, IEEE, Wen-Hua Chen, Fellow, IEEE, Jun Yang, Fellow, IEEE +Yunda Yan, Member, IEEE +Abstract—This paper develops a dual control framework for +exploration and exploitation (DCEE) to solve a self-optimisation +problem in unknown and uncertain environment. In general, +there is a fundamental conflict between tracking an unknown +optimal operational condition and parameter identification. Dif- +ferent from existing adaptive control methods, the proposed +DCEE does not need to introduce additional perturbation signals, +since it naturally embraces an exploration effect to actively +probe the uncertain environment to reduce belief uncertainty. An +ensemble based multi-estimator approach is developed to learn +the environmental parameters and in the meanwhile quantify the +estimation uncertainty in real time. The control action is devised +with dual effects, which not only minimises the tracking error +between the current state and the believed unknown optimal +operational condition but also reduces belief uncertainty by +actively exploring the environment. Formal properties of the +proposed DCEE framework like convergence are established. A +numerical example is used to validate the effectiveness of the +proposed DCEE. Simulation results for maximum power point +tracking are provided to further demonstrate the potential of +this new framework in real world applications. +Index Terms—Dual control, self-optimisation control, active +learning, exploration and exploitation, adaptation and control. +I. INTRODUCTION +Traditionally, adaptive control algorithms are mostly de- +signed for either regulation problems with known setpoints or +tracking problems with known reference trajectories. In many +applications, setpoints or references are usually dependent +on unknown or changing environment parameters, and thus +cannot be pre-specified in advance. Operating a system at +optimal condition is strongly desirable for best profit, pro- +ductivity or efficiency, but it can be particularly challenging +in an unknown or changing environment due to the presence +of uncertainties, disturbances and noises. Typical examples +include anti-lock braking systems to maintain maximal friction +under various unknown road surfaces and vehicle conditions +This work was supported by the UK Engineering and Physical Sciences +Research Council (EPSRC) Established Career Fellowship “Goal-Oriented +Control Systems: Disturbance, Uncertainty and Constraints” under the grant +number EP/T005734/1. +Z. Li is with Department of Computer Science, University College London, +London, WC1E 6BT, U.K. (email: zhongguo.li@ucl.ac.uk). +W.-H. Chen and J. Yang are with Department of Aeronautical and Automo- +tive Engineering, Loughborough University, Loughborough, LE11 3TU, U.K. +(emails: w.chen@lboro.ac.uk; j.yang3@lboro.ac.uk). +Y. +Yan +is +with +School +of +Engineering +and +Sustainable +Devel- +opment, +De +Montfort +University, +Leicester, +LE1 +9BH, +U.K. +(email: +yunda.yan@dmu.ac.uk). +[1], maximum power point tracking to continuously deliver the +highest possible power to the load in presence of variations in +environments [2], [3]. +As a classic control problem with a wide range of applica- +tions, early solution for static optimal operation can be traced +as far back as 1922 [4], [5]. It was popular in 1950s and 1960s, +and regained significant attention since 2000s due to a solid +theoretical foundation established for the stability and per- +formance in [6], [7]. Several approaches have been proposed +under different names including self-optimisation control [8], +extremum seeking control [6], [9] and hill-climbing systems +[10]. The goal of self-optimisation control is to keep the +system operating at a setpoint that optimises a performance +function dependent upon unknown or changing environment +parameters, despite uncertainties, disturbances and noises. +Since the optimal operation is unknown and possibly changes +during the operation, a control system must be able to adapt +to unknown or changing environments, for example, by means +of learning, adaptation and action through limited interactions +between the system and its operational environment. Then, +the control system devises possible strategies to track the +estimated setpoints or references based on its perceived en- +vironment knowledge and the level of confidence. +Generally speaking, there are dual objectives in a self- +optimisation control problem in an unknown and uncertain +environment: parameter identification and optimality tracking. +Quite often, the dual objectives are conflicting in the sense +that new observations do not provide sufficient information +for identifying the unknown parameters when the system state +settles to some local optimal solutions. This phenomenon +widely exists in adaptive extremum seeking when an extreme +searching algorithm converges to its local optimal solution, the +identifiability will naturally loss due to the lack of persistent +excitation (PE). As a trade-off, dither perturbations are intro- +duced on purpose to sustain the identifiability, but such dithers +inevitably deteriorate the tracking performance. Various ap- +proaches have been proposed to design the dither signals, e.g., +sinusoidal perturbations [6], [11], stochastic perturbations [12], +[13] and decaying perturbations [14]. However, they are usu- +ally pre-specified, and thereby cannot make online adjustments +according to real-time inference performance. In other words, +active learning cannot be embedded, that is, actively generate +data for the purpose of learning. +This paper proposes a new approach to self-optimisation +control by embedding active learning from a new perspective: + +2 +dual control of exploration and exploitation (DCEE). DCEE +was originally proposed in [15] for autonomous search of +sources of atmospheric release where the source location +and other environmental factors are unknown. To realise +autonomous search, it proposes each move of the robotic +agent shall have dual effects: driving the agent towards the +believed location of the source (exploitation) and probing the +environment to reduce the level of uncertainty of the current +belief (exploration). An optimal autonomous search strategy is +realised by optimally trading-off these two effects. We argue +in this paper that DCEE is actually applicable to a much wider +range of systems that operate in an unknown or uncertain +environment without well-defined control specifications (e.g. +the reward or cost functions are unknown). We present a new +self-optimisation control framework by extending DCEE from +a specific autonomous search application to a general design +approach for achieving or maintaining optimal operation in an +unknown environment. +The contribution of this paper is of twofold. On one side, +for self-optimisation control problems, we propose a new +and systematic framework which is able to actively probe +the environment to reduce the level of uncertainty through +active learning. There is no need to artificially introduce +perturbation as in the current extremum seeking control. It +also provides an optimal transition from any initial operation +condition to acquire the unknown optimal operation condition +in terms of a reformulated objective conditional upon current +knowledge and future predicted information. By formulating +the self-optimisation control in this framework, it enables to +establish proven properties by getting access to a wide range of +theoretic tools in control theory such as parameter adaptation +and optimal control. On the other side, we generalise and +extend the DCEE concept from a specific application, where +specific system dynamics, reward function and properties are +considered, to a general control system problem. A systematic +design procedure for general descriptions of the system and +control objectives is presented. We show that DCEE provides a +powerful and promising framework to design control systems +operating in an uncertain environment, which is an important +feature of autonomous systems. +Compared +with +all +the +existing +schemes +for +self- +optimisation control, our approach is most related to the work +where the model based approach is adopted and the uncertainty +of the objective or system dynamics are parameterised by +uncertain parameters [6], [8], [9], [16], [17]. There are three +main features in the new DCEE based self-optimisation control +framework, detailed as follows. +1) It is developed through an optimal control approach, +which is able to achieve best transition from any admis- +sible initial operation condition to the optimal operation +condition in terms of a reformulated objective. +2) It embeds an active learning effect allowing the system +to actively explore the unknown environment to reduce +the level of uncertainty. Instead of using computationally +expensive particle filters in information-driven methods, +this paper develops an efficient multi-estimator based en- +semble approach to quantify the estimation uncertainty +online, based on which the controller effectively trades +off between exploration and exploitation to balance the +dual objectives of identification and tracking. +3) Different from all the existing schemes where probing +effect is artificially introduced or inserted (usually by +means of dithers and perturbations), the probing effect +naturally occurs depending on the confidence of the es- +timation by assembling the outcomes of these individual +estimators. +The rest of this paper is organised as follows. In Sec- +tion II, we formulate the self-optimisation control problem and +demonstrate the dual effects embedded in the new formulation. +In Section III, an active learning based ensemble approach is +developed for unknown environment acquisition and then a +dual controller for exploration and exploitation is designed +to achieve optimal trade-off between parameter identification +and optimality tracking for a special single integrator system. +Section IV extends DCEE to general linear systems and formal +properties of the proposed self-optimisation control method are +established. Section V demonstrates the effectiveness of the +proposed algorithm using a numerical example. Section VI +formulates maximum power point tracking (MPPT) problem +as a self-optimisation control problem and compares the pro- +posed algorithm with other existing approaches. Section VII +concludes this paper. +II. PROBLEM STATEMENT +In this section, we elaborate the dual effects embedded +in the reformulated self-optimisation control problem. Then, +an ensemble active learning based approach is introduced to +realise efficient parameter adaptation and assess the estimation +performance. +A. Dual Control Reformulation +Consider a reward function for a system operating in an +unknown environment +J(θ∗, y) = φT(y)θ∗ +(1) +where θ∗ ⊂ Rm is unknown, depending on the operational +environment, y ∈ Rq is the system output, and φ(y) ∈ Rm +is the basis function of the reward function. In other words, +the reward function is parameterised by unknown θ∗. Without +loss of generality, it is assumed the the optimal condition is +achieved at the maximum of J. A self-optimisation control +is designed to automatically drive the system to the unknown +operational condition, maintain there despite disturbances and +automatically adjust the optimal operation condition accord- +ingly when the operational environment changes. +The system dynamics under concern are described by +x(k + 1) = Ax(k) + Bu(k) +y(k) = Cx(k) +(2) +where x(k) ∈ Rn, u(k) ∈ Rp and y(k) ∈ Rq are system state, +control input and output, respectively, and A ∈ Rn×n, B ∈ +Rn×p, C ∈ Rq×n are constant matrices. Suppose that at each +time, the system output and the reward J(k) can be measured +or derived subject to measurement noise v(k). We have +z(k) = [x(k); y(k); J(k) + v(k)] +(3) + +3 +and the information state is denoted as +Ik = [u(k − 1); z(k)] +(4) +All the measurement up to the current time k is given by +Ik = [I0, I1, . . . , Ik] +(5) +with I0 = [z(0)]. +There are two ways to formulate this problem using the dual +control for exploration and exploitation (DCEE) concept. The +first approach is similar to extremum seeking control [9], [16] +aiming to select the control such that the reward function is +maximised with all the information up to now including the +prior and all the measurements +max +u(k)∈Rp Eθ,Ik+1|k{J(θ, y(k + 1|k))|Ik+1|k} +(6) +subject to the system dynamics (2), where Ik+1|k += +[Ik, Ik+1|k] with Ik+1|k = [u(k), z(k + 1|k)]. z(k + 1|k) +consists of the predicted output y(k + 1|k) and the predicted +reward function under the control u(k). +Another approach is to drive the system output to the +unknown optimal condition directly, which is closer to the +classic self-optimisation control [8]. Since the optimal oper- +ation condition is unknown, the best one can do is to drive +the system to the best estimation of the optimal operation +condition with all the information up to now. This can be +formulated as +min +u(k)∈Rp E{(y(k + 1|k) − r∗)T(y(k + 1|k) − r∗)|Ik+1|k} (7) +where r∗ = l(θ∗) denotes the predicted optimal operational +condition conditional upon Ik+1|k, which is a function of +the environment parameter θ∗. In realm of self-optimisation +control, it is often required that the mapping l(θ) is a smooth +function of θ and r∗ = l(θ∗) is a unique optimum of the +objective function [6]. +These two problems have been solved previously in au- +tonomous search [15], [18]. The research question is how to +extend these results from this specific application to general +self-optimisation control problems. In this paper, we will focus +our attention on the latter formulation in (7), which is related to +the operational condition determined by unknown environment +parameters. +Before proceeding further, we demonstrate that the control +input u(k) obtained by minimising (7) naturally carries dual +effects corresponding to exploration and exploitation, respec- +tively. Intuitively, the control input u(k) will influence the +future system state y(k+1|k) via the system dynamics (2), and +at the same time affect the future information to be collected +Ik+1|k via the reward function in (1) from the environment +subject to uncertainties. +We define the predicted nominal operational condition as +¯r(k + 1|k) = E +� +r∗(k + 1|k)|Ik+1|k +� +(8) +based on which the prediction error conditional on Ik+1|k can +be written as +˜r(k + 1|k) = r∗(k + 1|k) − ¯r(k + 1|k). +(9) +Expanding (7) and substituting (8) and (9) into (7), we have +E +� +∥y(k + 1|k) − ¯r(k + 1|k) − ˜r(k + 1|k)∥2|Ik+1|k +� += E +� +∥y(k + 1|k) − ¯r(k + 1|k)∥2|Ik+1|k +� +− 2 E +� +(y(k + 1|k) − ¯r(k + 1|k))T˜r(k + 1|k)|Ik+1|k +� ++ E +� +∥˜r(k + 1|k)∥2|Ik+1|k +� +. +(10) +It follows from the definition of ˜r(k + 1|k) in (9) that +E +� +˜r(k + 1|k)|Ik+1|k +� += 0. Thus, by further noting that +y(k + 1|k) and ¯r(k + 1|k) are deterministic, the cross term in +(10) equals to zero, yielding +D(u(k)) := E +� +∥y(k + 1|k) − ¯r(k + 1|k)∥2|Ik+1|k +� ++ E +� +∥˜r(k + 1|k)∥2|Ik+1|k +� +. +(11) +Remark 1: The objective function in (11) exhibits dual +effects. Minimising the first term in (11) drives the system +state to estimated nominal value, which corresponds to the +exploitation effect. In control terminology, it can be understood +as tracking a nominal reference, thus also referred to as +optimality tracking. The second term characterises the level +of uncertainty (variance) associated with the predicted optimal +operational condition, which is related to the exploration +effect. According to the classic dual control concept [19], +[20], a control input is said to have dual effects if it can +affect at least one rth-order central moment of a state variable +(r > 1), in addition to its effect on the state. In fact, +the dual control framework developed in this paper is a +generalisation of the classic one [19] in the sense that our +formulation deals with not only system uncertainty but also +environment uncertainty (the operational condition r∗ = l(θ∗) +is determined by the environment parameters θ∗). This subtle +difference endows the system with capability of exploring the +operational environment and in the meanwhile exploiting its +current belief. Recently, DCEE has demonstrated superior and +promising performance in autonomous search [15], [21]. +Remark 2: According to [22], the level of autonomy can be +measured in terms of the set of goals that the system is able to +accomplish subject to a set of uncertainties. As a result, it is +required that the system can exploit its available knowledge to +accomplish the goals, and at the same time it should be able to +actively explore the operational environment to reduce knowl- +edge uncertainty. Effective trading-off between exploration and +exploitation has been a long standing issue, particularly in +artificial intelligence, control and decision-making in complex +and uncertain environment. In control society, some recent +works explicitly introduce trade-off coefficients to incorporate +the exploration terms into model predictive control problems, +e.g., [17], [23]. This inevitably incurs tedious efforts in tuning +the coefficients to balance exploration and exploitation. In +view of the derivation of (11), it is clear that the dual effects +in DCEE are naturally embedded, since they are derived from +a physically meaningful value function in (7). +B. Ensemble based Active Learning +Efficient gradient descent algorithms can be used to es- +timate the unknown parameters. The performance of single +estimator based optimisation algorithm is quite poor, due to + +4 +noisy measurement and nonlinear modelling (see examples in +autonomous search [18], [24]). Recently, the ensemble-based +approximation in machine learning community has demon- +strated great success with tractable computational load [25], +[26]. In this paper, we develop a multi-estimator based learning +method for parameter adaptation, which shows comparable +performance as particle filter using much less computational +resources in autonomous search application [18]. +Considering an ensemble of N estimators, the dual formu- +lation in (11) becomes +min +u(k)∈Rp D(u) = ∥y(k + 1|k) − ¯r(k + 1|k)∥2 + P(k + 1|k) +subject to x(k + 1|k) = Ax(k) + Bu(k) +y(k + 1|k) = Cx(k + 1|k) +(12) +where the nominal estimate and variance of the estimated +optimal condition are drawn from the ensemble, i.e., +¯r(k + 1|k) = 1 +N +N +� +i=1 +ri(k + 1|k) = 1 +N +N +� +i=1 +l(θi(k + 1|k)) +(13) +P(k + 1|k) = 1 +N +N +� +i=1 +(ri(k + 1|k) − ¯ri(k + 1|k))T +× (ri(k + 1|k) − ¯ri(k + 1|k)) +(14) +where the subscript i ∈ N denotes the index of the estimators, +with N representing the set of the ensemble. Note that +the relationship between the predicted optimal condition and +the unknown parameter, i.e., ri +k+1|k = l(θi +k+1|k), is usually +known. For example, in autonomous search application, θ∗ +is composed of the unknown source location and other envi- +ronment parameters, like wind direction and wind speed. The +optimal operation condition r∗ in autonomous search is the +source location, i.e., part of θ∗, which serves as a tracking +reference for the search agent. +In order to estimate the unknown parameter θ∗, we apply a +gradient-descent regression method [27], designed as +θi(k) =θi(k − 1) − ηiφ(y(k − 1)) +× +� +φ(y(k − 1))Tθi(k − 1) − J(k − 1) +� +, ∀i ∈ N. +(15) +where ηi > 0 is the learning rate of the ith estimator; J(k−1) +denotes the observed reward with measurement noise in (3) at +y(k − 1); and θ(k) denotes the estimate of unknown reward +parameter θ∗. The estimators are randomly initialised or they +can be initialised according to a priori pdfs of the unknown +parameters if available. Denote the estimation error as ˜θi(k) = +θi(k) − θ∗. Then, by noting J(k − 1) = φ(y(k − 1))Tθ∗ + +v(k − 1), we have +˜θi(k) = +� +Im − ηiφ(y(k − 1))φ(y(k − 1))T� ˜θi(k − 1) +− ηiφ(y(k − 1))v(k), ∀i ∈ N. +(16) +Denoting +the +extended +parameter +error +as +˜Θ(k) += +col{˜θ1(k), . . . , ˜θN(k)}, where col{·} denotes a column vector +formed by stacking the elements on top of each other, (16) +can be written in a compact form as +˜Θ(k) = +� +IN ⊗ +� +Im − ηiφ(y(k − 1))φ(y(k − 1))T� �˜Θ(k − 1) +− +� +IN ⊗ ηiφ(y(k − 1)) +� +(1N ⊗ v(k − 1)). +(17) +In an ensemble-based adaptation, we take their average +as the current estimation of the unknown parameters. Thus, +averaging (16), we have +˜Θav(k) = 1 +N (1T +N ⊗ Im)˜Θ(k) += 1 +N (1T +N ⊗ Im) +� +IN ⊗ (Im − ηφφT) +� ˜Θ(k − 1) +− 1 +N (1T +N ⊗ Im) +� +IN ⊗ ηφ +� +(1N ⊗ v(k − 1)) += 1 +N (1T +N ⊗ Im)˜Θ(k − 1) − 1 +N (1T +N ⊗ ηφφT)˜Θ(k − 1) +− ηφv(k − 1). +(18) +Remark 3: An important observation is that even though we +have the same regressor φ at one time instant, its excitation im- +pact to each estimator will be different since φφT˜θi ̸= φφT˜θj, +∀i ̸= j, almost surely. Due to the introduction of parameter +extension by multiple estimators, at any time instant the aver- +age estimation can always be excited when there are sufficient +estimators. In addition, by introducing a group of estimators, +it is possible to evaluate and make full use of the estimation +uncertainty by sampling the outcomes of the ensemble in an +online manner, which is proved to be crucial in DCEE [15] +as we will discuss in the sequel. Another desirable feature of +the ensemble approach is its resilience to measurement noises. +In view of the last term in (18), instantaneous noises will be +averaged out under multiple estimators such that the overall +performance of the ensemble can be improved. +III. DCEE FOR SINGLE INTEGRATOR +A. Algorithm Development +In high-level decision-making, system behaviours are usu- +ally simplified as single integrators by ignoring low-level +dynamics. In this paper, we begin with DCEE for this special +case +y(k + 1) = y(k) + u(k). +(19) +For general linear systems, we will use this as an internal +reference generator, as will be shown later in Section IV. +With the estimated environment parameter in (15), the dual +controller can be designed as +y(k + 1) = y(k) + u(k) +u(k) = −δk +� +∇yC(k + 1|k) + ∇yP(k + 1|k) +� +(20) +where C(k + 1|k) = ∥y(k) − ¯r(k + 1|k)∥2 denotes the +exploitation term, and P(k+1|k) is the exploration term in the +dual objective (12). To obtain the future mean and covariance, +we utilise the classic principles in extended Kalman filter. +According to the gradient-descent regression in (15), the + +5 +predicted mean of the N ensemble θi(k + 1|k), denoted as +¯θ(k + 1|k), is given by +¯θ(k + 1|k) = 1 +N +N +� +i=1 +θi(k + 1|k) += 1 +N +N +� +i=1 +(1m − ηiFi(k + 1|k))Tθi(k) +(21) +where +Fi(k + 1|k) =[J(θi(k), y) − J(k + 1|k)]φ(y) +(22) +with J(k + 1|k) being the predicted future reward based +on current belief {θi(k), ∀i ∈ N}. Note that the predicted +future reward is noise-free as there is no influence from +sensory devices in prediction. In this paper, we use the average +of θi(k), ∀i ∈ N to evaluate the predicted future reward. +Similarly, the predicted variance of the ensemble is given by +P(k + 1|k) = trace(F (k + 1|k)TP(k|k)F (k + 1|k)) +(23) +where +F (k + 1|k) = col{F1(k + 1|k), . . . , FN(k + 1|k)} +P(k|k) = cov{θi(k), ∀i ∈ N} += diag{(θ1(k) − ¯θ(k))(θ1(k) − ¯θ(k))T, . . . , +(θN(k) − ¯θ(k))(θN(k) − ¯θ(k))T} +(24) +where cov{·} is a covariance operator evaluating the covari- +ance matrix of the ensemble, and diag{·} denotes a block- +diagonal matrix by putting the elements on its main diagonal. +Using the predicted mean ¯θi(k + 1|k) and the predicted co- +variance P(k+1|k) of the unknown environmental parameter, +the dual control terms in (2) can be obtained by using the +mapping between the operational condition and the unknown +environmental parameter, i.e., r = l(θ). +B. Convergence Analysis +In this section, we will examine the convergence of the +proposed dual control algorithm, by leveraging parameter +adaptation and optimisation techniques. To this end, we in- +troduce some fundamental assumptions that will be used to +facilitate the convergence analysis of the proposed dual control +algorithm. +Assumption 1: There exist positive constants T ∈ Z+ and +β > 0 such that +t+T +� +k=t +[φ(y(k))][φ(y(k))]T ≥ βIm > 0, ∀t > 0. +(25) +Assumption 2: The measurement noise v(k) is independent +and identically distributed with bounded variance, i.e., +E [v(k)] = 0 +E +� +∥v(k)∥2� +≤ ̺2. +(26) +Assumption 3: The reward function J(θ, y) is twice dif- +ferentiable and strictly concave on y for any θ ∈ Rm, that +is, +∂2J(θ, y) +∂y2 +> 0. +(27) +Remark 4: Assumption 1 is a standard persistent excitation +(PE) condition to ensure the identifiability of the unknown +environmental parameter θ. Extensive techniques on parameter +adaptation have been reported in the past few decades aiming +at relaxing or fulfilling the conditions of PE [9], [27]. If we +introduce a memory-based regressor extension to the param- +eter adaptation algorithm in (15), the PE condition can be +relaxed to interval excitation [27]. Assumption 2 implies that +the noises imposed on sensory information are unbiased with +bounded variances. Assumption 3 guarantees the existence +and uniqueness of the optimal operational condition, i.e., +r∗ = l(θ∗), which is widely used in adaptive self-optimisation +and extremum seeking control [9], [28]. Note that the mapping +between the optimal operational condition and parameter θ can +be obtained by solving ∂J(θ,y) +∂y += 0. +First, we examine the convergence of the gradient-descent +regression method in (15). +Theorem 1: Under Assumptions 1 and 2, there exists a +constant η∗ > 0 such that, for any 0 < ηi < η∗, the estimates, +ˆθi(k), ∀i ∈ N, converge to a bounded neighbourhood of the +true environmental parameter θ∗. Moreover, the mean-square- +error of the estimator is convergent and bounded by +E ∥˜θi(k)∥2 ≤ +η2 +i L2̺2 +1 − maxj∈{1,...,k−1} ∥Ai(j)∥ +(28) +where Ai(j) = Im − ηi[φ(y(j))][φ(y(j))]T and L denotes the +bound of the regressor φ. Moreover, in absence of measure- +ment noises, limk→∞ E ∥˜θi(k)∥2 = 0. +Proof: In view of (16) and Assumption 2, the expectation of +the estimate is given by +E[˜θi(k)] = +� +Im − ηi[φ(y(k − 1))][φ(y(k − 1))]T� +E[˜θi(k − 1)] +∀i ∈ N. +(29) +According to Assumption 1, there exists a constant η∗ such +that, for any 0 < ηi < η∗, 0 < ηi[φ(y(k−1))][φ(y(k−1))]T < +Im. Consequently, for any 0 < ηi < η∗, we have +0 < Im − ηi[φ(y(k − 1))][φ(y(k − 1))]T < Im. +(30) +It follows from (29) that +∥ E[˜θi(k)]∥ ≤ +��Im − ηi[φ(y(k − 1))][φ(y(k − 1))]T�� +× ∥ E[˜θi(k − 1)]∥, ∀i ∈ N. +(31) +Therefore, +∥ E[˜θi(k)]∥ ≤ +k +� +j=1 +��Im − ηi[φ(y(j − 1))][φ(y(j − 1))]T�� +× ∥ E[˜θi(0)]∥, ∀i ∈ N. +(32) +For any bounded error ˜θi(0), the expectation of the estimator +converge to zero. + +6 +Moreover, the variance of the estimators can be bounded +under Assumption 2. Taking the squared Euclidean norm of +(16) yields +∥˜θi(k)∥2 = +��� +Im − ηi[φ(y(k − 1))][φ(y(k − 1))]T�˜θi(k − 1) +��2 ++ ∥ηiφ(y(k − 1))v(k)∥2 +− 2 +� +[Im − ηi[φ(y(k − 1))][φ(y(k − 1))]T] +× ˜θi(k − 1) +� +[ηiφ(y(k − 1))v(k)] , ∀i ∈ N. +(33) +Applying expectation operation to (33) leads to +E ∥˜θi(k)∥2 = E +�� � +Im − ηi[φ(y(k − 1))][φ(y(k − 1))]T� +× ˜θi(k − 1) +��2 ++ E ∥ηiφ(y(k − 1))v(k)∥2, ∀i ∈ N. +(34) +where E [v(k)] = 0 has been used to eliminate the cross term. +Denoting Ai(k−1) = Im −ηi[φ(y(k−1))][φ(y(k−1))]T and +applying the variance bound in (26), we have +E ∥˜θi(k)∥2 ≤ E +��˜θi(k − 1) +��2 +Ai(k−1) + η2 +i L2̺2. +(35) +For any 0 < ηi < η∗, the mean-square-error of the estimator +is convergent and bounded by +E ∥˜θi(k)∥2 ≤ +η2 +i L2̺2 +1 − maxj∈{1,...,k−1} ∥Ai(j)∥. +(36) +In absence of measurement noise v(k) += +0, limk→∞ +E ∥˜θi(k)∥2 = 0. This completes the proof. +■ +Remark 5: Theorem 1 establishes the convergence of the +estimators under mild assumptions on the measurement noises +and persistent excitation. The parameter adaptation algorithm +together with its convergence analysis under measurement +noises forms a new feature of this paper since existing studies +mainly focus on noise-free scenarios [27], [29]. As having +been discussed in Remark 4, PE is a standard and commonly- +used condition to guarantee the convergence of parameter +estimators. Despite significant research efforts have been dedi- +cated to explore weak/alternative assumptions, very few result +has been obtained (see recent survey in [27]). In the proposed +dual controller (20), a probing effort is inherently embedded +aiming to reduce the estimation uncertainty. Such an explo- +ration effect from active learning is beneficial to environment +acquisition, which has been validated in autonomous search +application [15], [18]. +Remark 6: The proposed multi-estimator assisted ensemble +method for environment adaptation is a hybrid approach that +combines both model-based and model-free techniques. The +model-based estimators are trained according to the model +structures of the reward function in (1). A model-free ensemble +approximation is used to estimate the mean and variance of the +unknown environmental parameters in an online manner. It is +widely perceived in machine learning community that model- +based approach benefits from high learning efficiency due +to the utilisation of model knowledge but inevitably inherits +model biased errors; on the other hand, model-free approach +provides a reliable way to quantify the level of estimation +uncertainty but may incur additional computational burden. +Recently, the hybrid method has demonstrated superior perfor- +mance in simulation and experiment in machine learning due +to its combined strength from both model-based and model- +free learning [25], [26]. Theoretical guarantee on convergence +and performance of the hybrid approach has not been well- +established but mainly verified by extensive simulation and +experimental results. Inspired by its recent success, we develop +a concurrent active learning based ensemble algorithm and +establish its formal properties in this paper. +Denote the tracking error between current state and un- +known optimal condition r∗ as ˜y(k) = y(k) − r∗. Then, it +follows from (20) that +˜y(k + 1) = ˜y(k) − δk +� +∇yC(k + 1|k) + ∇yP(k + 1|k) +� +. +(37) +Now, we analyse the convergence to the optimal operational +condition. +Theorem 2: Under Assumptions 1-3, for any 0 < ηi < η∗, +y converges to a bounded neighbourhood of the optimal +operational condition r∗ = l(θ∗) if there exists a step size +δk such that 0 < 2∥[In − δkL(k)]∥2 < 1 with L(k) = +� 1 +0 ∇2 +yC(r∗ + τ ˜y(k))dτ. +Proof: To relate the gradient term ∇yC(k + 1|k) with ˜y(k), +we recall the mean value theorem [30], that is, for a twice- +differentiable function h(y) : Rm → R, +∇h(y1) =∇h(y2) + +�� 1 +0 +∇2h[y2 + τ(y1 − y2)]dτ +� +(y1 − y2), +∀y1, y2 ∈ Rm . +(38) +Thus, we have +∇yC(y(k)) =∇yC(r∗) + +� � 1 +0 +∇2 +yC(r∗ + τ ˜y(k))dτ +� +˜y(k) +(39) +where the time stamps in C(k+1|k) have been dropped for no- +tational convenience. Denoting L(k) = +� 1 +0 ∇2 +yC(r∗+τ ˜y(k))dτ +and applying ∇yC(r∗) = 0, we have +∇yC(y(k)) = L(k)˜y(k). +(40) +Applying (40) to (37) results in +˜y(k + 1) = [In − δkL(k)]˜y(k) − δk∇yP(k + 1|k). +(41) +To examine the boundedness of the tracking error, we take the +Euclidean norm for both sides of (41), yielding +∥˜y(k + 1)∥2 =∥[In − δkL(k)]˜y(k)∥2 + ∥δk∇yP(k + 1|k)∥2 +− 2δk[(In − δkL(k))˜y(k)]T∇T +yP(k + 1|k). +(42) +Taking the expectation of (42) leads to +E ∥˜y(k + 1)∥2 ≤∥[In − δkL(k)]∥2 E ∥˜y(k)∥2 ++ E ∥δk∇yP(k + 1|k)∥2 ++ E[−2δk∇T +yP(k + 1|k)[(In − δkL(k))˜y(k)]]. +(43) +The last term in (43) can be written as +E[−2δk∇T +yP(k + 1|k)[(In − δkL(k))˜y(k)]] +≤ ∥[In − δkL(k)]∥2 E ∥˜y(k)∥2 + E ∥δk∇yP(k + 1|k)∥2. +(44) + +7 +Therefore, substituting (44) into (43) results in +E ∥˜y(k + 1)∥2 ≤2∥[In − δkL(k)]∥2 E ∥˜y(k)∥2 ++ 2 E ∥δk∇yP(k + 1|k)∥2. +(45) +From Theorem 1, the estimation errors are bounded within +E ∥˜θi(k)∥2 ≤ max +� +∥˜θi(0)∥2, +η2 +i L2̺2 +1 − maxj∈{1,...,k−1} ρ(Ai(j)) +� +. +(46) +As a result, 0 ≤ E ∥δk∇yP(k +1|k)∥2 ≤ µ is upper bounded, +since it is a measure of covariance of the bounded estimators. +Consequently, we have +E ∥˜y(k + 1)∥2 ≤2∥[In − δkL(k)]∥2 E ∥˜y(k)∥2 + µ. +(47) +If there exists a step size δk such that 0 < 2∥[In−δkL(k)]∥2 < +1, then the expected mean square of the tracking error is +convergent. Recursively iterating (47) gives +E ∥˜y(k + 1)∥2 ≤ ¯αk E ∥˜y(0)∥2 + +k−1 +� +j=0 +¯αjµ +(48) +where ¯α := maxj∈{1,...,k} αj with 0 < αk := 2∥[In − +δjL(j)]∥2 < 1. Since limk→∞ ¯αk E ∥˜y(0)∥2 → 0, we have +lim +k→∞ E ∥y(k) − r∗∥2 ≤ +µ +1 − ¯α. +(49) +This completes the proof. +■ +Remark 7: In general, traditional adaptive control can +be regarded as passive learning [9], [17] where parameter +estimators are updated by accidentally collected data sam- +ples. For example, MPC in autonomous search is targeted at +navigating the agent to the source position, whereas during +this pure exploitation process the estimators are updated pas- +sively by accidentally collected concentration measurements +from the environment [15], [31]. Recently, there are a wide +range of engineering problems involved in balancing between +exploration and exploitation, e.g., machine learning, control +and decision-making in uncertain environment [20], [32]– +[34]. In control society, related works are usually focused on +stochastic model predictive control with active learning [17]. A +similar concept is referred to as active reinforcement learning +in artificial intelligence [34], [35]. Nevertheless, there is a +critical distinction between previous works and the proposed +DCEE framework for self-optimisation control. In existing +dual control formulation, the probing effect is introduced to +learn the system states or parameters (e.g. MPC with active +learning [36] and active adaptive control [37], [38]), while +in our formulation the probing effect is used to actively +explore the operational environment. We believe that future +autonomous control should be able to deal with not only +system uncertainty but also environment uncertainty [15], [22]. +IV. DCEE FOR LINEAR SYSTEMS +In this section, we deal with general linear systems. As +the environment estimators are designed by information mea- +surements, the parameter adaptation algorithm in (15) can be +used and Theorem 1 remains valid. Now, we design a dual +controller that regulates the system output y(k) to minimise +the reformulated objective function defined in (12). +The dual controller is proposed as +u(k) = −Kx(k) + (G + KΨ)ξ(k) +(50) +where the optimal reference ξ(k) is generated by +ξ(k) = ξ(k − 1) + ψ(k) +ψ(k) = −δk +� +∇ξC(k + 1|k) + ∇ξP(k + 1|k) +� +(51) +where G and Ψ are gain matrices obtained by solving +(A − I)Ψ + BG = 0 +CΨ − I = 0. +(52) +and K is chosen such that A − BK is Schur stable as (A, B) +is controllable. Note that ψ(k) is exactly the dual gradient +term used in the integrator dynamics in Section III. For +linear systems, the control input u(k) not only needs to have +dual effects for exploration and exploitation but additionally +requires control effort to stabilise the system dynamics as in +(50). +Assumption 4: The pair (A, B) is controllable, and +rank +� +A − I +B +C +0 +� += n + q. +(53) +Remark 8: The dual control design in (50)-(52) is partly +inspired by conventional internal model approaches [39]. The +solvability of (52) is guaranteed by (53), which is widely +known as regulation equations [39]. The existence of Ψ +ensures the existence of optimal state x∗ = Ψr∗ such that +Cx∗ = r∗. +Define state transformations xs(k) = Ψξ(k), us(k) = +Gξ(k). Let ¯x(k) = x(k) − xs(k) and ¯u(k) = u(k) − us(k). +Applying the transformation to the system dynamics (2) leads +to +¯x(k + 1) = x(k + 1) − xs(k + 1) += Ax(k) + Bu(k) − Ψ(ξ(k) + ψ(k)) += A¯x(k) + B¯u(k) − Ψψ(k) +e(k) = C¯x(k) +(54) +where (52) has been used to derive above dynamics. Applying +the control input (50), we have the closed loop dynamics +¯x(k + 1) = (A − BK)¯x(k) − Ψψ(k) +e(k) = C¯x(k). +(55) +The following lemma can be regarded as input-to-output +stability of the transformed dynamics (55) by viewing ψ(k) +and e(k) as the input and output, respectively. +Lemma 1: Let Assumptions 1–4 hold. Suppose the condi- +tions specified in Theorems 1–2 hold. If the gain matrices G +and Ψ are designed according to (52) and K is chosen such +that (A − BK) is Schur stable, then +lim sup +k→∞ +∥e(k)∥ ≤ +1 +1 − ∥A − BK∥ lim sup +k→∞ +∥ψ(k)∥ +(56) +Furthermore, if lim supk→∞ ψ(k) = 0, then lim supk→∞ e(k) += 0. +Proof: Putting (52) into a matrix form leads to +� A − I +B +C +0 +� � Ψ +G +� += +� 0 +I +� +(57) + +8 +of which the solvability is guaranteed under (53) in Assump- +tion 4 by transforming the matrix equation (57) to standard +linear algebraic equations. For notational convenience, we +denote Ac = A − BK and Bc = −Ψ. Then, we have +¯x(k + 1) = Ac¯x(k) + Bcψ(k). +(58) +Recursively iterating (58) results in +¯x(k) = Ak +c ¯x(0) + +k−1 +� +j=0 +Ak−j−1 +c +Bcψ(j). +(59) +Hence, we have +e(k) = C¯x(k) = CAk +c ¯x(0) − +k−1 +� +j=0 +Ak−j−1 +c +ψ(j) +(60) +where CΨ − I = 0 has been used. Because Ac is Schur, we +have limk→∞ CAk +c ¯x(0) = 0. +The convergence of reference generator (51) has been +established in Theorem 2, and thereby ψ(k), i.e., the gradient +of the dual controller, is bounded and converges to zero as +k → ∞. Denoting ̟ := lim supk→∞ ∥ψ(k)∥, it can be +obtained that, for any small constant ǫ > 0, there exists a +positive time index ζ > 0 such that +∥ψ(k)∥ < ̟ + ǫ, ∀k > ζ. +(61) +Now, the second term in (60) can be separated into two +parts, written as +k−1 +� +j=0 +Ak−j−1 +c +ψ(j) = +ζ +� +j=0 +Ak−j−1 +c +ψ(j) + +k−1 +� +j=ζ+1 +Ak−j−1 +c +ψ(j). +(62) +Taking the Euclidean norm of (62) and invoking (61), we +obtain +���� +k−1 +� +j=0 +Ak−j−1 +c +ψ(j) +���� = +���� +ζ +� +j=0 +Ak−j−1 +c +ψ(j) ++ +k−1 +� +j=ζ+1 +Ak−j−1 +c +ψ(j) +���� +≤ +��Ak−ζ−1 +c +�� +���� +ζ +� +j=0 +Aζ−j +c +ψ(j) +���� + (̟ + ǫ) +���� +k−1 +� +j=ζ+1 +Ak−j−1 +c +����. +(63) +Therefore, combining (60) and (63) leads to +lim sup +k→∞ +∥e(k)∥ ≤ +1 +1 − ∥Ac∥ (̟ + ε) +(64) +by noting that +t−1 +� +j=ζ+1 +∥Ac∥t−1−j = 1 − ∥Ac∥t−ζ +1 − ∥Ac∥ +< +1 +1 − ∥Ac∥ +(65) +and that +lim +k→∞ +��Ak−ζ−1 +c +�� = 0 +(66) +since Ac is Schur stable. As ǫ can be set arbitrarily small, it +follows from (64) that +lim sup +k→∞ +∥e(k)∥ ≤ +1 +1 − ∥Ac∥ lim sup +k→∞ +∥ψ(k)∥. +(67) +This completes the proof. +■ +Now, combining the results in Theorems 1–2 and Lemma 1, +we are ready to establish the convergence of the self- +optimisation control for linear systems. +Theorem 3: Let Assumptions 1–4 hold. Suppose the +conditions specified in Theorems 1–2 and Lemma 1 hold. +The output y(k) of the linear system (2) converges to the +neighbourhood of the optimum r∗, using control input (50) to- +gether with reference generator (51). Moreover, in the absence +of measurement noises, y(k) converges to the true optimal +solution r∗. +Proof: Denoting ˜x(k) = x(k) − Ψr∗, we have +˜x(k + 1) = Ax(k) + B[−Kx(k) + (G + KΨ)ξ(k)] − Ψr∗ += (A − BK)˜x(k) + B(G + KΨ)(ξ(k) − r∗) +(68) +It follows from Theorems 1–2 that ξ(k) converges to the +neighbourhood of r∗ with bounded error. Thus, the result can +be concluded by treating B(G + KΨ)(ξ(k) − r∗) as ψ(k) in +Lemma 1. +■ +Remark 9: The self-optimisation control in this paper is +similar to the classic formulation of reinforcement learning +in the sense that both of them are targeted to operate a +system in an unknown and uncertain environment. There are +two bottlenecks in widely applying reinforcement learning, +particularly deep RL: one is a large number of trials are +required to achieve a satisfactory performance (big data) and +the other is its performance could significantly degrade if +the real operational environment is different from the training +environment (poor adaptiveness) [40]. DCEE establishes a new +control framework to provide a promising and complementary +method to reinforcement learning in control and robotics +society. In fact, active learning for exploration and exploitation +in machine intelligence can find strong evidence in human +intelligence, which is supported by the biological principles +in functional integration in the human brain and neuronal in- +teractions (known as free-energy principle and active inference +in neuroscience [41]). Interested readers are referred to [40] +for detailed discussions. +V. NUMERICAL EXAMPLE +In this section, we verify the effectiveness of the proposed +algorithm using a dedicate numerical example. Consider a +linear system (2) with +A = +� +0 +1 +2 +1 +� +, B = +� +1 +1 +� +, C = +� 0 +1 � +. +(69) +The reward function is given by +J(θ∗, y) = 2y − θ∗y2 = +� 2y +−y2 � � 1 +θ∗ +� +(70) +where θ∗ is affected by the unknown environment. The true +value is θ∗ = 1 but unavailable a priori. The optimal oper- +ational condition r∗ is determined by θ∗, i.e., r∗ = l(θ∗) = +1/θ∗ = 1. +We assume the measurements are subject to Gaussian noise +v(k) ∼ N(0, 2), which implies that the observations from +environment are J(k) = J(θ∗, y(k))+ v(k). Decision-making + +9 +under uncertain environment with noisy measurements is of +significant importance to promote the system intelligence. +In order to explore the uncertain environment, the first step +is to quantify the level of uncertainty. An ensemble based +multi-estimator approach has been developed in previous +sections. Now, the size of the estimator ensemble is chosen +as N += 100, and each of them is randomly initialised +according to a uniform distribution between 0 and 20, i.e., +θi(0) ∼ U(0, 20), ∀i = 1, 2, . . . , 100. The step sizes are set +as ηi = 0.005 and δk = 0.5. The system is controllable +and regulation condition in (53) is satisfied such that the gain +matrices can be obtained as Ψ = [ 1 +3, 1]T and G = − 2 +3. The +gain matrix K = [−1.24, 1.14] is chosen by placing the poles +of (A − BK) at [0.4; 0.7]. +Fig. 1 shows the estimated environmental parameters. Ini- +tially, the mean and standard deviation of the ensemble +{θi, i = 1, . . . , 100} are 10.87 and 5.57, respectively, ran- +domly initialised using a uniform distribution. The mean of +the estimators converges to the true environment parameter +θ∗ = 1, and the standard deviation among the estimators +shrinks quickly, indicating that the estimation uncertainty +reduces (quantified by the variance among the estimators in +the ensemble). Despite increasing the iteration k significantly, +the estimated parameters remain fluctuating within a small +neighbourhood of the true value due to the presence of noisy +measurements. Fig. 2 displays the observed rewards from the +environment. Even though we have imposed quite significant +noises to the measurements, the performance of the estimators +is fairly satisfactory, which manifests the ensemble based +active learning provides superior robustness against noises. +Implementing the dual control in (50) not only contributes to +enhanced parameter adaptation performance but also drives the +system output to the optimal operational condition, as shown in +Fig. 3. The system output approaches the optimal operational +point r∗ = 1 as shown in Fig. 3, and the system states are +displayed in Fig. 4. It can be verified that x∗ = Ψr∗ = [ 1 +3, 1]T. +The tracking error is determined by the estimation error. In this +process, there is no need to tune the weights of exploration +and exploitation. As a principled approach, the dual controller +in (50) is derived from a physically meaningful objective +function, which naturally embeds balanced dual effects for +active environment learning and optimality tracking. +VI. APPLICATION FOR MPPT +DCEE was originally developed to solve autonomous search +problem in [15], which demonstrates outstanding performance +compared with other existing approaches. In this section, we +take the optimal control for photovoltaic (PV) systems as an +example to illustrate that DCEE can be implemented to solve +a much wider class of self-optimisation control problems in +real-world applications. Extracting maximum power is a long- +lasting pursuit in operating PV systems. Despite significant +research efforts made over the past few decades [42]–[44], +the energy conversion efficiency of PV systems remains very +poor due to high environment uncertainties in temperature, +irradiance level, partial shading and other atmospheric con- +ditions. The primary goal in PV operation is simply to +Fig. 1: Mean and standard deviation of estimated θ(k) using +ensemble based estimators. +Fig. 2: Observed reward J(k) from unknown and uncertain +environment with measurement noises v(t). +extract solar energy as much as possible despite changing +operational environment, termed as maximum power point +tracking (MPPT). There have been a wide variety of methods +targeting to solve this problem, which can be roughly classified +into three categories: offline methods, online methods, and +other methods. Detailed comparisons and classifications can +be found in comprehensive survey papers, e.g., [42], [43]. +In this section, the proposed DCEE is implemented as an +alternative approach to achieve MPPT, and two representa- +tive approaches, hill climbing method (HC) and incremental +conductance method (IC), are deployed for comparison. It is +Fig. 3: System output y(k) using DCEE. + +10 +Fig. 4: System state x(k). +Fig. 5: Time-varying solar irradiance profile. +worth noting that all the three algorithms can be classified +as online methods. It has been widely perceived that online +methods usually outperform offline counterparts in terms of +conversion efficiency due to their inherent adaptiveness to +changing environment. According to the curve-fitting based +MPPT [42], the power and voltage (P-V ) characteristics can +be modelled by +P = φT(V )θ +(71) +where φ(V ) is the polynomial regressor [1, V, V 2, . . . , V n]T +and θ ∈ Rn+1 is the polynomial coefficient. To solve the +maximum problem of (71), we need to estimate the unknown +parameters θ and then maximise the power output by regulat- +ing the voltage V according to +V (k + 1) = V (k) + u(k). +(72) +We use solar panel A10Green Technology model number +A10J-S72-175 for this simulation [45]. To mimic the real +operational environment of PV systems, a time-varying solar +irradiance profile is stimulated as shown in Fig. 5, and the +temperature is initially set as 25°C and then jumps to 35°C +at t = 1s. It should be noted that the unknown environment +parameter θ changes as the operational condition varies. Al- +though the proposed algorithm is theoretically analysed for +static parameters identification, the use of constant learning +rate ηi renders the adaptation algorithm in (15) with the +capability of tracking drifting parameters. +Simulation results using different algorithms (DCEE, HC +and IC) are shown in Fig. 6, 7 and 8. To illustrate more de- +tailed features of different algorithms, enlarged sub-figures are +displayed for the time intervals t ∈ [0, 0.1], and t ∈ [0.3, 0.4]. +The power losses, as displayed in Fig. 9, are calculated by +integrating the differences between the maximum power point +and real power outputs stimulated using different algorithms. +Convergence speed, sensed signals, algorithm complexity and +conversion efficiency are four commonly-used criteria to as- +sess the characteristics of MPPT techniques. According to the +simulation results, we summarise and compare the features of +different approaches in Table I. Conversion efficiency directly +influences the energy extracted from the PV systems, which +is ratio between real generated energy and maximum energy +(accumulated over the simulation time interval [0, 2]). DCEE +produces quite high efficiency (99.1%). Due to the use of +perturbed signals in hill climbing method, there are very +large voltage and current fluctuations in steady state. This +undesirable property not only causes low conversion efficiency +but also leads to fast degradation in low level electronic +devices. The oscillations are partially solved by incremental +conductance method, which measures incremental current and +voltage changes to predict the effect of voltage change. +Different from HC, incremental inductance method is able +to maintain at MPP without oscillations when there is no +change in operational environment. From the simulation re- +sults using HC and IC, there is a trade-off between transient +convergence speed and steady-state oscillations. The steady- +state oscillation of IC is reduced at the cost of slow tracking +performance, leading to larger power loss with a conversion +efficiency 97.2%. It is argued that DCEE as a balanced ap- +proach is able to optimally trade-off between exploitation and +exploration: when there is large uncertainty in estimated MPP, +it will explore quickly to gain information to construct more +accurate estimate of MPP; and when there is less change in +operational environment, it will maintain at the current belief +of MPP without causing large oscillations. All three algorithms +need to measure voltage and current: DCEE requires voltage +and power (calculated by the product of current and voltage) to +construct P-V curve in (71) (i.e., reward-state mapping), while +HC and IC use incremental power to deicide the direction of +voltage regulation. As mature MPPT techniques, both HC and +IC are simple to implement using dedicated hardware devices. +Since efficient ensemble approximation and gradient based +control are developed in this new approach, DCEE is ready to +be implemented in real PV platforms without incurring heavy +computational load. +VII. CONCLUSION +In this paper, a general framework of dual control for ex- +ploration and exploitation has been developed to solve a wide +range of self-optimisation control problems in an uncertain +environment. A real-time ensemble based estimation approach +is proposed for efficient environment acquisition, which con- +sequently provides a measure of knowledge uncertainty to +the unknown environment. The proposed DCEE algorithm +optimally balances between exploration and exploitation to + +11 +TABLE I: Features of different MPPT techniques. +Methods +Convergence speed +Sensed variables +Algorithm complexity +Conversion efficiency +1 +DCEE +Fast +Voltage and current +Medium +99.1% +2 +Hill climbing +Fast +Voltage and current +Simple +98.3% +3 +Incremental conductance +Medium +Voltage and current +Simple +97.2% +Fig. 6: Power profile using different algorithms. +Fig. 7: Voltage profile using different algorithms. +Fig. 8: Current profile using different algorithms. +Fig. 9: Power losses using different algorithms. +handle the intrinsic conflict between parameter identifiability +and optimality tracking. Guaranteed convergence and perfor- +mance are established in relation to the reward function and +the noise characteristics. A numerical example and a classic +application of MPPT are provided to validate the effectiveness +and potential of DCEE. +REFERENCES +[1] C. Zhang and R. 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Tey, “Maximum power point tracking +using modified butterfly optimization algorithm for partial shading, +uniform shading, and fast varying load conditions,” IEEE Transactions +on Power Electronics, vol. 36, no. 5, pp. 5569–5581, 2020. + diff --git a/ANFLT4oBgHgl3EQfEi_Y/content/tmp_files/load_file.txt b/ANFLT4oBgHgl3EQfEi_Y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e7e7d370b29aa0f22b607a2622a9811e46197e62 --- /dev/null +++ b/ANFLT4oBgHgl3EQfEi_Y/content/tmp_files/load_file.txt @@ -0,0 +1,755 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf,len=754 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='11984v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='SY] 27 Jan 2023 1 Dual Control of Exploration and Exploitation for Self-Optimisation Control in Uncertain Environments Zhongguo Li, Member, IEEE, Wen-Hua Chen, Fellow, IEEE, Jun Yang, Fellow, IEEE Yunda Yan, Member, IEEE Abstract—This paper develops a dual control framework for exploration and exploitation (DCEE) to solve a self-optimisation problem in unknown and uncertain environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' In general, there is a fundamental conflict between tracking an unknown optimal operational condition and parameter identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Dif- ferent from existing adaptive control methods, the proposed DCEE does not need to introduce additional perturbation signals, since it naturally embraces an exploration effect to actively probe the uncertain environment to reduce belief uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' An ensemble based multi-estimator approach is developed to learn the environmental parameters and in the meanwhile quantify the estimation uncertainty in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The control action is devised with dual effects, which not only minimises the tracking error between the current state and the believed unknown optimal operational condition but also reduces belief uncertainty by actively exploring the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Formal properties of the proposed DCEE framework like convergence are established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' A numerical example is used to validate the effectiveness of the proposed DCEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Simulation results for maximum power point tracking are provided to further demonstrate the potential of this new framework in real world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Index Terms—Dual control, self-optimisation control, active learning, exploration and exploitation, adaptation and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' INTRODUCTION Traditionally, adaptive control algorithms are mostly de- signed for either regulation problems with known setpoints or tracking problems with known reference trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' In many applications, setpoints or references are usually dependent on unknown or changing environment parameters, and thus cannot be pre-specified in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Operating a system at optimal condition is strongly desirable for best profit, pro- ductivity or efficiency, but it can be particularly challenging in an unknown or changing environment due to the presence of uncertainties, disturbances and noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Typical examples include anti-lock braking systems to maintain maximal friction under various unknown road surfaces and vehicle conditions This work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) Established Career Fellowship “Goal-Oriented Control Systems: Disturbance, Uncertainty and Constraints” under the grant number EP/T005734/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Li is with Department of Computer Science, University College London, London, WC1E 6BT, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (email: zhongguo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='li@ucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Chen and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Yang are with Department of Aeronautical and Automo- tive Engineering, Loughborough University, Loughborough, LE11 3TU, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (emails: w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='chen@lboro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='uk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='yang3@lboro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Yan is with School of Engineering and Sustainable Devel- opment, De Montfort University, Leicester, LE1 9BH, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (email: yunda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='yan@dmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' [1], maximum power point tracking to continuously deliver the highest possible power to the load in presence of variations in environments [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' As a classic control problem with a wide range of applica- tions, early solution for static optimal operation can be traced as far back as 1922 [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' It was popular in 1950s and 1960s, and regained significant attention since 2000s due to a solid theoretical foundation established for the stability and per- formance in [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Several approaches have been proposed under different names including self-optimisation control [8], extremum seeking control [6], [9] and hill-climbing systems [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The goal of self-optimisation control is to keep the system operating at a setpoint that optimises a performance function dependent upon unknown or changing environment parameters, despite uncertainties, disturbances and noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Since the optimal operation is unknown and possibly changes during the operation, a control system must be able to adapt to unknown or changing environments, for example, by means of learning, adaptation and action through limited interactions between the system and its operational environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Then, the control system devises possible strategies to track the estimated setpoints or references based on its perceived en- vironment knowledge and the level of confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Generally speaking, there are dual objectives in a self- optimisation control problem in an unknown and uncertain environment: parameter identification and optimality tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Quite often, the dual objectives are conflicting in the sense that new observations do not provide sufficient information for identifying the unknown parameters when the system state settles to some local optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' This phenomenon widely exists in adaptive extremum seeking when an extreme searching algorithm converges to its local optimal solution, the identifiability will naturally loss due to the lack of persistent excitation (PE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' As a trade-off, dither perturbations are intro- duced on purpose to sustain the identifiability, but such dithers inevitably deteriorate the tracking performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Various ap- proaches have been proposed to design the dither signals, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=', sinusoidal perturbations [6], [11], stochastic perturbations [12], [13] and decaying perturbations [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' However, they are usu- ally pre-specified, and thereby cannot make online adjustments according to real-time inference performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' In other words, active learning cannot be embedded, that is, actively generate data for the purpose of learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' This paper proposes a new approach to self-optimisation control by embedding active learning from a new perspective: 2 dual control of exploration and exploitation (DCEE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' DCEE was originally proposed in [15] for autonomous search of sources of atmospheric release where the source location and other environmental factors are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' To realise autonomous search, it proposes each move of the robotic agent shall have dual effects: driving the agent towards the believed location of the source (exploitation) and probing the environment to reduce the level of uncertainty of the current belief (exploration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' An optimal autonomous search strategy is realised by optimally trading-off these two effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' We argue in this paper that DCEE is actually applicable to a much wider range of systems that operate in an unknown or uncertain environment without well-defined control specifications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' the reward or cost functions are unknown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' We present a new self-optimisation control framework by extending DCEE from a specific autonomous search application to a general design approach for achieving or maintaining optimal operation in an unknown environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The contribution of this paper is of twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' On one side, for self-optimisation control problems, we propose a new and systematic framework which is able to actively probe the environment to reduce the level of uncertainty through active learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' There is no need to artificially introduce perturbation as in the current extremum seeking control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' It also provides an optimal transition from any initial operation condition to acquire the unknown optimal operation condition in terms of a reformulated objective conditional upon current knowledge and future predicted information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' By formulating the self-optimisation control in this framework, it enables to establish proven properties by getting access to a wide range of theoretic tools in control theory such as parameter adaptation and optimal control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' On the other side, we generalise and extend the DCEE concept from a specific application, where specific system dynamics, reward function and properties are considered, to a general control system problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' A systematic design procedure for general descriptions of the system and control objectives is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' We show that DCEE provides a powerful and promising framework to design control systems operating in an uncertain environment, which is an important feature of autonomous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Compared with all the existing schemes for self- optimisation control, our approach is most related to the work where the model based approach is adopted and the uncertainty of the objective or system dynamics are parameterised by uncertain parameters [6], [8], [9], [16], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' There are three main features in the new DCEE based self-optimisation control framework, detailed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' 1) It is developed through an optimal control approach, which is able to achieve best transition from any admis- sible initial operation condition to the optimal operation condition in terms of a reformulated objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' 2) It embeds an active learning effect allowing the system to actively explore the unknown environment to reduce the level of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Instead of using computationally expensive particle filters in information-driven methods, this paper develops an efficient multi-estimator based en- semble approach to quantify the estimation uncertainty online, based on which the controller effectively trades off between exploration and exploitation to balance the dual objectives of identification and tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' 3) Different from all the existing schemes where probing effect is artificially introduced or inserted (usually by means of dithers and perturbations), the probing effect naturally occurs depending on the confidence of the es- timation by assembling the outcomes of these individual estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The rest of this paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' In Sec- tion II, we formulate the self-optimisation control problem and demonstrate the dual effects embedded in the new formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' In Section III, an active learning based ensemble approach is developed for unknown environment acquisition and then a dual controller for exploration and exploitation is designed to achieve optimal trade-off between parameter identification and optimality tracking for a special single integrator system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Section IV extends DCEE to general linear systems and formal properties of the proposed self-optimisation control method are established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Section V demonstrates the effectiveness of the proposed algorithm using a numerical example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Section VI formulates maximum power point tracking (MPPT) problem as a self-optimisation control problem and compares the pro- posed algorithm with other existing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Section VII concludes this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' PROBLEM STATEMENT In this section, we elaborate the dual effects embedded in the reformulated self-optimisation control problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Then, an ensemble active learning based approach is introduced to realise efficient parameter adaptation and assess the estimation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Dual Control Reformulation Consider a reward function for a system operating in an unknown environment J(θ∗, y) = φT(y)θ∗ (1) where θ∗ ⊂ Rm is unknown, depending on the operational environment, y ∈ Rq is the system output, and φ(y) ∈ Rm is the basis function of the reward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' In other words, the reward function is parameterised by unknown θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Without loss of generality, it is assumed the the optimal condition is achieved at the maximum of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' A self-optimisation control is designed to automatically drive the system to the unknown operational condition, maintain there despite disturbances and automatically adjust the optimal operation condition accord- ingly when the operational environment changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The system dynamics under concern are described by x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k) (2) where x(k) ∈ Rn, u(k) ∈ Rp and y(k) ∈ Rq are system state, control input and output, respectively, and A ∈ Rn×n, B ∈ Rn×p, C ∈ Rq×n are constant matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Suppose that at each time, the system output and the reward J(k) can be measured or derived subject to measurement noise v(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' We have z(k) = [x(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' y(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' J(k) + v(k)] (3) 3 and the information state is denoted as Ik = [u(k − 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' z(k)] (4) All the measurement up to the current time k is given by Ik = [I0, I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' , Ik] (5) with I0 = [z(0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' There are two ways to formulate this problem using the dual control for exploration and exploitation (DCEE) concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The first approach is similar to extremum seeking control [9], [16] aiming to select the control such that the reward function is maximised with all the information up to now including the prior and all the measurements max u(k)∈Rp Eθ,Ik+1|k{J(θ, y(k + 1|k))|Ik+1|k} (6) subject to the system dynamics (2), where Ik+1|k = [Ik, Ik+1|k] with Ik+1|k = [u(k), z(k + 1|k)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' z(k + 1|k) consists of the predicted output y(k + 1|k) and the predicted reward function under the control u(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Another approach is to drive the system output to the unknown optimal condition directly, which is closer to the classic self-optimisation control [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Since the optimal oper- ation condition is unknown, the best one can do is to drive the system to the best estimation of the optimal operation condition with all the information up to now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' This can be formulated as min u(k)∈Rp E{(y(k + 1|k) − r∗)T(y(k + 1|k) − r∗)|Ik+1|k} (7) where r∗ = l(θ∗) denotes the predicted optimal operational condition conditional upon Ik+1|k, which is a function of the environment parameter θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' In realm of self-optimisation control, it is often required that the mapping l(θ) is a smooth function of θ and r∗ = l(θ∗) is a unique optimum of the objective function [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' These two problems have been solved previously in au- tonomous search [15], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The research question is how to extend these results from this specific application to general self-optimisation control problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' In this paper, we will focus our attention on the latter formulation in (7), which is related to the operational condition determined by unknown environment parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Before proceeding further, we demonstrate that the control input u(k) obtained by minimising (7) naturally carries dual effects corresponding to exploration and exploitation, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Intuitively, the control input u(k) will influence the future system state y(k+1|k) via the system dynamics (2), and at the same time affect the future information to be collected Ik+1|k via the reward function in (1) from the environment subject to uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' We define the predicted nominal operational condition as ¯r(k + 1|k) = E � r∗(k + 1|k)|Ik+1|k � (8) based on which the prediction error conditional on Ik+1|k can be written as ˜r(k + 1|k) = r∗(k + 1|k) − ¯r(k + 1|k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (9) Expanding (7) and substituting (8) and (9) into (7), we have E � ∥y(k + 1|k) − ¯r(k + 1|k) − ˜r(k + 1|k)∥2|Ik+1|k � = E � ∥y(k + 1|k) − ¯r(k + 1|k)∥2|Ik+1|k � − 2 E � (y(k + 1|k) − ¯r(k + 1|k))T˜r(k + 1|k)|Ik+1|k � + E � ∥˜r(k + 1|k)∥2|Ik+1|k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (10) It follows from the definition of ˜r(k + 1|k) in (9) that E � ˜r(k + 1|k)|Ik+1|k � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Thus, by further noting that y(k + 1|k) and ¯r(k + 1|k) are deterministic, the cross term in (10) equals to zero, yielding D(u(k)) := E � ∥y(k + 1|k) − ¯r(k + 1|k)∥2|Ik+1|k � + E � ∥˜r(k + 1|k)∥2|Ik+1|k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (11) Remark 1: The objective function in (11) exhibits dual effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Minimising the first term in (11) drives the system state to estimated nominal value, which corresponds to the exploitation effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' In control terminology, it can be understood as tracking a nominal reference, thus also referred to as optimality tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The second term characterises the level of uncertainty (variance) associated with the predicted optimal operational condition, which is related to the exploration effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' According to the classic dual control concept [19], [20], a control input is said to have dual effects if it can affect at least one rth-order central moment of a state variable (r > 1), in addition to its effect on the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' In fact, the dual control framework developed in this paper is a generalisation of the classic one [19] in the sense that our formulation deals with not only system uncertainty but also environment uncertainty (the operational condition r∗ = l(θ∗) is determined by the environment parameters θ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' This subtle difference endows the system with capability of exploring the operational environment and in the meanwhile exploiting its current belief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Recently, DCEE has demonstrated superior and promising performance in autonomous search [15], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Remark 2: According to [22], the level of autonomy can be measured in terms of the set of goals that the system is able to accomplish subject to a set of uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' As a result, it is required that the system can exploit its available knowledge to accomplish the goals, and at the same time it should be able to actively explore the operational environment to reduce knowl- edge uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Effective trading-off between exploration and exploitation has been a long standing issue, particularly in artificial intelligence, control and decision-making in complex and uncertain environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' In control society, some recent works explicitly introduce trade-off coefficients to incorporate the exploration terms into model predictive control problems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=', [17], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' This inevitably incurs tedious efforts in tuning the coefficients to balance exploration and exploitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' In view of the derivation of (11), it is clear that the dual effects in DCEE are naturally embedded, since they are derived from a physically meaningful value function in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Ensemble based Active Learning Efficient gradient descent algorithms can be used to es- timate the unknown parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The performance of single estimator based optimisation algorithm is quite poor, due to 4 noisy measurement and nonlinear modelling (see examples in autonomous search [18], [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Recently, the ensemble-based approximation in machine learning community has demon- strated great success with tractable computational load [25], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' In this paper, we develop a multi-estimator based learning method for parameter adaptation, which shows comparable performance as particle filter using much less computational resources in autonomous search application [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Considering an ensemble of N estimators, the dual formu- lation in (11) becomes min u(k)∈Rp D(u) = ∥y(k + 1|k) − ¯r(k + 1|k)∥2 + P(k + 1|k) subject to x(k + 1|k) = Ax(k) + Bu(k) y(k + 1|k) = Cx(k + 1|k) (12) where the nominal estimate and variance of the estimated optimal condition are drawn from the ensemble, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=', ¯r(k + 1|k) = 1 N N � i=1 ri(k + 1|k) = 1 N N � i=1 l(θi(k + 1|k)) (13) P(k + 1|k) = 1 N N � i=1 (ri(k + 1|k) − ¯ri(k + 1|k))T × (ri(k + 1|k) − ¯ri(k + 1|k)) (14) where the subscript i ∈ N denotes the index of the estimators, with N representing the set of the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Note that the relationship between the predicted optimal condition and the unknown parameter, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=', ri k+1|k = l(θi k+1|k), is usually known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' For example, in autonomous search application, θ∗ is composed of the unknown source location and other envi- ronment parameters, like wind direction and wind speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The optimal operation condition r∗ in autonomous search is the source location, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=', part of θ∗, which serves as a tracking reference for the search agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' In order to estimate the unknown parameter θ∗, we apply a gradient-descent regression method [27], designed as θi(k) =θi(k − 1) − ηiφ(y(k − 1)) × � φ(y(k − 1))Tθi(k − 1) − J(k − 1) � , ∀i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (15) where ηi > 0 is the learning rate of the ith estimator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' J(k−1) denotes the observed reward with measurement noise in (3) at y(k − 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' and θ(k) denotes the estimate of unknown reward parameter θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The estimators are randomly initialised or they can be initialised according to a priori pdfs of the unknown parameters if available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Denote the estimation error as ˜θi(k) = θi(k) − θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Then, by noting J(k − 1) = φ(y(k − 1))Tθ∗ + v(k − 1), we have ˜θi(k) = � Im − ηiφ(y(k − 1))φ(y(k − 1))T� ˜θi(k − 1) − ηiφ(y(k − 1))v(k), ∀i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (16) Denoting the extended parameter error as ˜Θ(k) = col{˜θ1(k), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' , ˜θN(k)}, where col{·} denotes a column vector formed by stacking the elements on top of each other, (16) can be written in a compact form as ˜Θ(k) = � IN ⊗ � Im − ηiφ(y(k − 1))φ(y(k − 1))T� �˜Θ(k − 1) − � IN ⊗ ηiφ(y(k − 1)) � (1N ⊗ v(k − 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (17) In an ensemble-based adaptation, we take their average as the current estimation of the unknown parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Thus, averaging (16), we have ˜Θav(k) = 1 N (1T N ⊗ Im)˜Θ(k) = 1 N (1T N ⊗ Im) � IN ⊗ (Im − ηφφT) � ˜Θ(k − 1) − 1 N (1T N ⊗ Im) � IN ⊗ ηφ � (1N ⊗ v(k − 1)) = 1 N (1T N ⊗ Im)˜Θ(k − 1) − 1 N (1T N ⊗ ηφφT)˜Θ(k − 1) − ηφv(k − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (18) Remark 3: An important observation is that even though we have the same regressor φ at one time instant, its excitation im- pact to each estimator will be different since φφT˜θi ̸= φφT˜θj, ∀i ̸= j, almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Due to the introduction of parameter extension by multiple estimators, at any time instant the aver- age estimation can always be excited when there are sufficient estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' In addition, by introducing a group of estimators, it is possible to evaluate and make full use of the estimation uncertainty by sampling the outcomes of the ensemble in an online manner, which is proved to be crucial in DCEE [15] as we will discuss in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Another desirable feature of the ensemble approach is its resilience to measurement noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' In view of the last term in (18), instantaneous noises will be averaged out under multiple estimators such that the overall performance of the ensemble can be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' DCEE FOR SINGLE INTEGRATOR A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Algorithm Development In high-level decision-making, system behaviours are usu- ally simplified as single integrators by ignoring low-level dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' In this paper, we begin with DCEE for this special case y(k + 1) = y(k) + u(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (19) For general linear systems, we will use this as an internal reference generator, as will be shown later in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' With the estimated environment parameter in (15), the dual controller can be designed as y(k + 1) = y(k) + u(k) u(k) = −δk � ∇yC(k + 1|k) + ∇yP(k + 1|k) � (20) where C(k + 1|k) = ∥y(k) − ¯r(k + 1|k)∥2 denotes the exploitation term, and P(k+1|k) is the exploration term in the dual objective (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' To obtain the future mean and covariance, we utilise the classic principles in extended Kalman filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' According to the gradient-descent regression in (15), the 5 predicted mean of the N ensemble θi(k + 1|k), denoted as ¯θ(k + 1|k), is given by ¯θ(k + 1|k) = 1 N N � i=1 θi(k + 1|k) = 1 N N � i=1 (1m − ηiFi(k + 1|k))Tθi(k) (21) where Fi(k + 1|k) =[J(θi(k), y) − J(k + 1|k)]φ(y) (22) with J(k + 1|k) being the predicted future reward based on current belief {θi(k), ∀i ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Note that the predicted future reward is noise-free as there is no influence from sensory devices in prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' In this paper, we use the average of θi(k), ∀i ∈ N to evaluate the predicted future reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Similarly, the predicted variance of the ensemble is given by P(k + 1|k) = trace(F (k + 1|k)TP(k|k)F (k + 1|k)) (23) where F (k + 1|k) = col{F1(k + 1|k), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' , FN(k + 1|k)} P(k|k) = cov{θi(k), ∀i ∈ N} = diag{(θ1(k) − ¯θ(k))(θ1(k) − ¯θ(k))T, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' , (θN(k) − ¯θ(k))(θN(k) − ¯θ(k))T} (24) where cov{·} is a covariance operator evaluating the covari- ance matrix of the ensemble, and diag{·} denotes a block- diagonal matrix by putting the elements on its main diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Using the predicted mean ¯θi(k + 1|k) and the predicted co- variance P(k+1|k) of the unknown environmental parameter, the dual control terms in (2) can be obtained by using the mapping between the operational condition and the unknown environmental parameter, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=', r = l(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Convergence Analysis In this section, we will examine the convergence of the proposed dual control algorithm, by leveraging parameter adaptation and optimisation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' To this end, we in- troduce some fundamental assumptions that will be used to facilitate the convergence analysis of the proposed dual control algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Assumption 1: There exist positive constants T ∈ Z+ and β > 0 such that t+T � k=t [φ(y(k))][φ(y(k))]T ≥ βIm > 0, ∀t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (25) Assumption 2: The measurement noise v(k) is independent and identically distributed with bounded variance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=', E [v(k)] = 0 E � ∥v(k)∥2� ≤ ̺2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (26) Assumption 3: The reward function J(θ, y) is twice dif- ferentiable and strictly concave on y for any θ ∈ Rm, that is, ∂2J(θ, y) ∂y2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (27) Remark 4: Assumption 1 is a standard persistent excitation (PE) condition to ensure the identifiability of the unknown environmental parameter θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Extensive techniques on parameter adaptation have been reported in the past few decades aiming at relaxing or fulfilling the conditions of PE [9], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' If we introduce a memory-based regressor extension to the param- eter adaptation algorithm in (15), the PE condition can be relaxed to interval excitation [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Assumption 2 implies that the noises imposed on sensory information are unbiased with bounded variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Assumption 3 guarantees the existence and uniqueness of the optimal operational condition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=', r∗ = l(θ∗), which is widely used in adaptive self-optimisation and extremum seeking control [9], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Note that the mapping between the optimal operational condition and parameter θ can be obtained by solving ∂J(θ,y) ∂y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' First, we examine the convergence of the gradient-descent regression method in (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Theorem 1: Under Assumptions 1 and 2, there exists a constant η∗ > 0 such that, for any 0 < ηi < η∗, the estimates, ˆθi(k), ∀i ∈ N, converge to a bounded neighbourhood of the true environmental parameter θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Moreover, the mean-square- error of the estimator is convergent and bounded by E ∥˜θi(k)∥2 ≤ η2 i L2̺2 1 − maxj∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=',k−1} ∥Ai(j)∥ (28) where Ai(j) = Im − ηi[φ(y(j))][φ(y(j))]T and L denotes the bound of the regressor φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Moreover, in absence of measure- ment noises, limk→∞ E ∥˜θi(k)∥2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Proof: In view of (16) and Assumption 2, the expectation of the estimate is given by E[˜θi(k)] = � Im − ηi[φ(y(k − 1))][φ(y(k − 1))]T� E[˜θi(k − 1)] ∀i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (29) According to Assumption 1, there exists a constant η∗ such that, for any 0 < ηi < η∗, 0 < ηi[φ(y(k−1))][φ(y(k−1))]T < Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Consequently, for any 0 < ηi < η∗, we have 0 < Im − ηi[φ(y(k − 1))][φ(y(k − 1))]T < Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (30) It follows from (29) that ∥ E[˜θi(k)]∥ ≤ ��Im − ηi[φ(y(k − 1))][φ(y(k − 1))]T�� × ∥ E[˜θi(k − 1)]∥, ∀i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (31) Therefore, ∥ E[˜θi(k)]∥ ≤ k � j=1 ��Im − ηi[φ(y(j − 1))][φ(y(j − 1))]T�� × ∥ E[˜θi(0)]∥, ∀i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (32) For any bounded error ˜θi(0), the expectation of the estimator converge to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' 6 Moreover, the variance of the estimators can be bounded under Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Taking the squared Euclidean norm of (16) yields ∥˜θi(k)∥2 = ��� Im − ηi[φ(y(k − 1))][φ(y(k − 1))]T�˜θi(k − 1) ��2 + ∥ηiφ(y(k − 1))v(k)∥2 − 2 � [Im − ηi[φ(y(k − 1))][φ(y(k − 1))]T] × ˜θi(k − 1) � [ηiφ(y(k − 1))v(k)] , ∀i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (33) Applying expectation operation to (33) leads to E ∥˜θi(k)∥2 = E �� � Im − ηi[φ(y(k − 1))][φ(y(k − 1))]T� × ˜θi(k − 1) ��2 + E ∥ηiφ(y(k − 1))v(k)∥2, ∀i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (34) where E [v(k)] = 0 has been used to eliminate the cross term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Denoting Ai(k−1) = Im −ηi[φ(y(k−1))][φ(y(k−1))]T and applying the variance bound in (26), we have E ∥˜θi(k)∥2 ≤ E ��˜θi(k − 1) ��2 Ai(k−1) + η2 i L2̺2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (35) For any 0 < ηi < η∗, the mean-square-error of the estimator is convergent and bounded by E ∥˜θi(k)∥2 ≤ η2 i L2̺2 1 − maxj∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=',k−1} ∥Ai(j)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (36) In absence of measurement noise v(k) = 0, limk→∞ E ∥˜θi(k)∥2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' ■ Remark 5: Theorem 1 establishes the convergence of the estimators under mild assumptions on the measurement noises and persistent excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The parameter adaptation algorithm together with its convergence analysis under measurement noises forms a new feature of this paper since existing studies mainly focus on noise-free scenarios [27], [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' As having been discussed in Remark 4, PE is a standard and commonly- used condition to guarantee the convergence of parameter estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Despite significant research efforts have been dedi- cated to explore weak/alternative assumptions, very few result has been obtained (see recent survey in [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' In the proposed dual controller (20), a probing effort is inherently embedded aiming to reduce the estimation uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Such an explo- ration effect from active learning is beneficial to environment acquisition, which has been validated in autonomous search application [15], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Remark 6: The proposed multi-estimator assisted ensemble method for environment adaptation is a hybrid approach that combines both model-based and model-free techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The model-based estimators are trained according to the model structures of the reward function in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' A model-free ensemble approximation is used to estimate the mean and variance of the unknown environmental parameters in an online manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' It is widely perceived in machine learning community that model- based approach benefits from high learning efficiency due to the utilisation of model knowledge but inevitably inherits model biased errors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' on the other hand, model-free approach provides a reliable way to quantify the level of estimation uncertainty but may incur additional computational burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Recently, the hybrid method has demonstrated superior perfor- mance in simulation and experiment in machine learning due to its combined strength from both model-based and model- free learning [25], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Theoretical guarantee on convergence and performance of the hybrid approach has not been well- established but mainly verified by extensive simulation and experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Inspired by its recent success, we develop a concurrent active learning based ensemble algorithm and establish its formal properties in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Denote the tracking error between current state and un- known optimal condition r∗ as ˜y(k) = y(k) − r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Then, it follows from (20) that ˜y(k + 1) = ˜y(k) − δk � ∇yC(k + 1|k) + ∇yP(k + 1|k) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (37) Now, we analyse the convergence to the optimal operational condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Theorem 2: Under Assumptions 1-3, for any 0 < ηi < η∗, y converges to a bounded neighbourhood of the optimal operational condition r∗ = l(θ∗) if there exists a step size δk such that 0 < 2∥[In − δkL(k)]∥2 < 1 with L(k) = � 1 0 ∇2 yC(r∗ + τ ˜y(k))dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Proof: To relate the gradient term ∇yC(k + 1|k) with ˜y(k), we recall the mean value theorem [30], that is, for a twice- differentiable function h(y) : Rm → R, ∇h(y1) =∇h(y2) + �� 1 0 ∇2h[y2 + τ(y1 − y2)]dτ � (y1 − y2), ∀y1, y2 ∈ Rm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (38) Thus, we have ∇yC(y(k)) =∇yC(r∗) + � � 1 0 ∇2 yC(r∗ + τ ˜y(k))dτ � ˜y(k) (39) where the time stamps in C(k+1|k) have been dropped for no- tational convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Denoting L(k) = � 1 0 ∇2 yC(r∗+τ ˜y(k))dτ and applying ∇yC(r∗) = 0, we have ∇yC(y(k)) = L(k)˜y(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (40) Applying (40) to (37) results in ˜y(k + 1) = [In − δkL(k)]˜y(k) − δk∇yP(k + 1|k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (41) To examine the boundedness of the tracking error, we take the Euclidean norm for both sides of (41), yielding ∥˜y(k + 1)∥2 =∥[In − δkL(k)]˜y(k)∥2 + ∥δk∇yP(k + 1|k)∥2 − 2δk[(In − δkL(k))˜y(k)]T∇T yP(k + 1|k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (42) Taking the expectation of (42) leads to E ∥˜y(k + 1)∥2 ≤∥[In − δkL(k)]∥2 E ∥˜y(k)∥2 + E ∥δk∇yP(k + 1|k)∥2 + E[−2δk∇T yP(k + 1|k)[(In − δkL(k))˜y(k)]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (43) The last term in (43) can be written as E[−2δk∇T yP(k + 1|k)[(In − δkL(k))˜y(k)]] ≤ ∥[In − δkL(k)]∥2 E ∥˜y(k)∥2 + E ∥δk∇yP(k + 1|k)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (44) 7 Therefore, substituting (44) into (43) results in E ∥˜y(k + 1)∥2 ≤2∥[In − δkL(k)]∥2 E ∥˜y(k)∥2 + 2 E ∥δk∇yP(k + 1|k)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (45) From Theorem 1, the estimation errors are bounded within E ∥˜θi(k)∥2 ≤ max � ∥˜θi(0)∥2, η2 i L2̺2 1 − maxj∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=',k−1} ρ(Ai(j)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (46) As a result, 0 ≤ E ∥δk∇yP(k +1|k)∥2 ≤ µ is upper bounded, since it is a measure of covariance of the bounded estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Consequently, we have E ∥˜y(k + 1)∥2 ≤2∥[In − δkL(k)]∥2 E ∥˜y(k)∥2 + µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (47) If there exists a step size δk such that 0 < 2∥[In−δkL(k)]∥2 < 1, then the expected mean square of the tracking error is convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Recursively iterating (47) gives E ∥˜y(k + 1)∥2 ≤ ¯αk E ∥˜y(0)∥2 + k−1 � j=0 ¯αjµ (48) where ¯α := maxj∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=',k} αj with 0 < αk := 2∥[In − δjL(j)]∥2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Since limk→∞ ¯αk E ∥˜y(0)∥2 → 0, we have lim k→∞ E ∥y(k) − r∗∥2 ≤ µ 1 − ¯α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (49) This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' ■ Remark 7: In general, traditional adaptive control can be regarded as passive learning [9], [17] where parameter estimators are updated by accidentally collected data sam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' For example, MPC in autonomous search is targeted at navigating the agent to the source position, whereas during this pure exploitation process the estimators are updated pas- sively by accidentally collected concentration measurements from the environment [15], [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Recently, there are a wide range of engineering problems involved in balancing between exploration and exploitation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=', machine learning, control and decision-making in uncertain environment [20], [32]– [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' In control society, related works are usually focused on stochastic model predictive control with active learning [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' A similar concept is referred to as active reinforcement learning in artificial intelligence [34], [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Nevertheless, there is a critical distinction between previous works and the proposed DCEE framework for self-optimisation control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' In existing dual control formulation, the probing effect is introduced to learn the system states or parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' MPC with active learning [36] and active adaptive control [37], [38]), while in our formulation the probing effect is used to actively explore the operational environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' We believe that future autonomous control should be able to deal with not only system uncertainty but also environment uncertainty [15], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' DCEE FOR LINEAR SYSTEMS In this section, we deal with general linear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' As the environment estimators are designed by information mea- surements, the parameter adaptation algorithm in (15) can be used and Theorem 1 remains valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Now, we design a dual controller that regulates the system output y(k) to minimise the reformulated objective function defined in (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The dual controller is proposed as u(k) = −Kx(k) + (G + KΨ)ξ(k) (50) where the optimal reference ξ(k) is generated by ξ(k) = ξ(k − 1) + ψ(k) ψ(k) = −δk � ∇ξC(k + 1|k) + ∇ξP(k + 1|k) � (51) where G and Ψ are gain matrices obtained by solving (A − I)Ψ + BG = 0 CΨ − I = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (52) and K is chosen such that A − BK is Schur stable as (A, B) is controllable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Note that ψ(k) is exactly the dual gradient term used in the integrator dynamics in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' For linear systems, the control input u(k) not only needs to have dual effects for exploration and exploitation but additionally requires control effort to stabilise the system dynamics as in (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Assumption 4: The pair (A, B) is controllable, and rank � A − I B C 0 � = n + q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (53) Remark 8: The dual control design in (50)-(52) is partly inspired by conventional internal model approaches [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The solvability of (52) is guaranteed by (53), which is widely known as regulation equations [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The existence of Ψ ensures the existence of optimal state x∗ = Ψr∗ such that Cx∗ = r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Define state transformations xs(k) = Ψξ(k), us(k) = Gξ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Let ¯x(k) = x(k) − xs(k) and ¯u(k) = u(k) − us(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Applying the transformation to the system dynamics (2) leads to ¯x(k + 1) = x(k + 1) − xs(k + 1) = Ax(k) + Bu(k) − Ψ(ξ(k) + ψ(k)) = A¯x(k) + B¯u(k) − Ψψ(k) e(k) = C¯x(k) (54) where (52) has been used to derive above dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Applying the control input (50), we have the closed loop dynamics ¯x(k + 1) = (A − BK)¯x(k) − Ψψ(k) e(k) = C¯x(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (55) The following lemma can be regarded as input-to-output stability of the transformed dynamics (55) by viewing ψ(k) and e(k) as the input and output, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Lemma 1: Let Assumptions 1–4 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Suppose the condi- tions specified in Theorems 1–2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' If the gain matrices G and Ψ are designed according to (52) and K is chosen such that (A − BK) is Schur stable, then lim sup k→∞ ∥e(k)∥ ≤ 1 1 − ∥A − BK∥ lim sup k→∞ ∥ψ(k)∥ (56) Furthermore, if lim supk→∞ ψ(k) = 0, then lim supk→∞ e(k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Proof: Putting (52) into a matrix form leads to � A − I B C 0 � � Ψ G � = � 0 I � (57) 8 of which the solvability is guaranteed under (53) in Assump- tion 4 by transforming the matrix equation (57) to standard linear algebraic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' For notational convenience, we denote Ac = A − BK and Bc = −Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Then, we have ¯x(k + 1) = Ac¯x(k) + Bcψ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (58) Recursively iterating (58) results in ¯x(k) = Ak c ¯x(0) + k−1 � j=0 Ak−j−1 c Bcψ(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (59) Hence, we have e(k) = C¯x(k) = CAk c ¯x(0) − k−1 � j=0 Ak−j−1 c ψ(j) (60) where CΨ − I = 0 has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Because Ac is Schur, we have limk→∞ CAk c ¯x(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The convergence of reference generator (51) has been established in Theorem 2, and thereby ψ(k), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=', the gradient of the dual controller, is bounded and converges to zero as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Denoting ̟ := lim supk→∞ ∥ψ(k)∥, it can be obtained that, for any small constant ǫ > 0, there exists a positive time index ζ > 0 such that ∥ψ(k)∥ < ̟ + ǫ, ∀k > ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (61) Now, the second term in (60) can be separated into two parts, written as k−1 � j=0 Ak−j−1 c ψ(j) = ζ � j=0 Ak−j−1 c ψ(j) + k−1 � j=ζ+1 Ak−j−1 c ψ(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (62) Taking the Euclidean norm of (62) and invoking (61), we obtain ���� k−1 � j=0 Ak−j−1 c ψ(j) ���� = ���� ζ � j=0 Ak−j−1 c ψ(j) + k−1 � j=ζ+1 Ak−j−1 c ψ(j) ���� ≤ ��Ak−ζ−1 c �� ���� ζ � j=0 Aζ−j c ψ(j) ���� + (̟ + ǫ) ���� k−1 � j=ζ+1 Ak−j−1 c ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (63) Therefore, combining (60) and (63) leads to lim sup k→∞ ∥e(k)∥ ≤ 1 1 − ∥Ac∥ (̟ + ε) (64) by noting that t−1 � j=ζ+1 ∥Ac∥t−1−j = 1 − ∥Ac∥t−ζ 1 − ∥Ac∥ < 1 1 − ∥Ac∥ (65) and that lim k→∞ ��Ak−ζ−1 c �� = 0 (66) since Ac is Schur stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' As ǫ can be set arbitrarily small, it follows from (64) that lim sup k→∞ ∥e(k)∥ ≤ 1 1 − ∥Ac∥ lim sup k→∞ ∥ψ(k)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (67) This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' ■ Now, combining the results in Theorems 1–2 and Lemma 1, we are ready to establish the convergence of the self- optimisation control for linear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Theorem 3: Let Assumptions 1–4 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Suppose the conditions specified in Theorems 1–2 and Lemma 1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The output y(k) of the linear system (2) converges to the neighbourhood of the optimum r∗, using control input (50) to- gether with reference generator (51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Moreover, in the absence of measurement noises, y(k) converges to the true optimal solution r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Proof: Denoting ˜x(k) = x(k) − Ψr∗, we have ˜x(k + 1) = Ax(k) + B[−Kx(k) + (G + KΨ)ξ(k)] − Ψr∗ = (A − BK)˜x(k) + B(G + KΨ)(ξ(k) − r∗) (68) It follows from Theorems 1–2 that ξ(k) converges to the neighbourhood of r∗ with bounded error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Thus, the result can be concluded by treating B(G + KΨ)(ξ(k) − r∗) as ψ(k) in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' ■ Remark 9: The self-optimisation control in this paper is similar to the classic formulation of reinforcement learning in the sense that both of them are targeted to operate a system in an unknown and uncertain environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' There are two bottlenecks in widely applying reinforcement learning, particularly deep RL: one is a large number of trials are required to achieve a satisfactory performance (big data) and the other is its performance could significantly degrade if the real operational environment is different from the training environment (poor adaptiveness) [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' DCEE establishes a new control framework to provide a promising and complementary method to reinforcement learning in control and robotics society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' In fact, active learning for exploration and exploitation in machine intelligence can find strong evidence in human intelligence, which is supported by the biological principles in functional integration in the human brain and neuronal in- teractions (known as free-energy principle and active inference in neuroscience [41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Interested readers are referred to [40] for detailed discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' NUMERICAL EXAMPLE In this section, we verify the effectiveness of the proposed algorithm using a dedicate numerical example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Consider a linear system (2) with A = � 0 1 2 1 � , B = � 1 1 � , C = � 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (69) The reward function is given by J(θ∗, y) = 2y − θ∗y2 = � 2y −y2 � � 1 θ∗ � (70) where θ∗ is affected by the unknown environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The true value is θ∗ = 1 but unavailable a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The optimal oper- ational condition r∗ is determined by θ∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=', r∗ = l(θ∗) = 1/θ∗ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' We assume the measurements are subject to Gaussian noise v(k) ∼ N(0, 2), which implies that the observations from environment are J(k) = J(θ∗, y(k))+ v(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Decision-making 9 under uncertain environment with noisy measurements is of significant importance to promote the system intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' In order to explore the uncertain environment, the first step is to quantify the level of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' An ensemble based multi-estimator approach has been developed in previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Now, the size of the estimator ensemble is chosen as N = 100, and each of them is randomly initialised according to a uniform distribution between 0 and 20, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=', θi(0) ∼ U(0, 20), ∀i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' , 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The step sizes are set as ηi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='005 and δk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The system is controllable and regulation condition in (53) is satisfied such that the gain matrices can be obtained as Ψ = [ 1 3, 1]T and G = − 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The gain matrix K = [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='24, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='14] is chosen by placing the poles of (A − BK) at [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' 1 shows the estimated environmental parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Ini- tially, the mean and standard deviation of the ensemble {θi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' , 100} are 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='87 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='57, respectively, ran- domly initialised using a uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The mean of the estimators converges to the true environment parameter θ∗ = 1, and the standard deviation among the estimators shrinks quickly, indicating that the estimation uncertainty reduces (quantified by the variance among the estimators in the ensemble).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Despite increasing the iteration k significantly, the estimated parameters remain fluctuating within a small neighbourhood of the true value due to the presence of noisy measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' 2 displays the observed rewards from the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Even though we have imposed quite significant noises to the measurements, the performance of the estimators is fairly satisfactory, which manifests the ensemble based active learning provides superior robustness against noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Implementing the dual control in (50) not only contributes to enhanced parameter adaptation performance but also drives the system output to the optimal operational condition, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The system output approaches the optimal operational point r∗ = 1 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' 3, and the system states are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' It can be verified that x∗ = Ψr∗ = [ 1 3, 1]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The tracking error is determined by the estimation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' In this process, there is no need to tune the weights of exploration and exploitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' As a principled approach, the dual controller in (50) is derived from a physically meaningful objective function, which naturally embeds balanced dual effects for active environment learning and optimality tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' APPLICATION FOR MPPT DCEE was originally developed to solve autonomous search problem in [15], which demonstrates outstanding performance compared with other existing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' In this section, we take the optimal control for photovoltaic (PV) systems as an example to illustrate that DCEE can be implemented to solve a much wider class of self-optimisation control problems in real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Extracting maximum power is a long- lasting pursuit in operating PV systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Despite significant research efforts made over the past few decades [42]–[44], the energy conversion efficiency of PV systems remains very poor due to high environment uncertainties in temperature, irradiance level, partial shading and other atmospheric con- ditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The primary goal in PV operation is simply to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' 1: Mean and standard deviation of estimated θ(k) using ensemble based estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' 2: Observed reward J(k) from unknown and uncertain environment with measurement noises v(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' extract solar energy as much as possible despite changing operational environment, termed as maximum power point tracking (MPPT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' There have been a wide variety of methods targeting to solve this problem, which can be roughly classified into three categories: offline methods, online methods, and other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Detailed comparisons and classifications can be found in comprehensive survey papers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=', [42], [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' In this section, the proposed DCEE is implemented as an alternative approach to achieve MPPT, and two representa- tive approaches, hill climbing method (HC) and incremental conductance method (IC), are deployed for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' It is Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' 3: System output y(k) using DCEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' 10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' 4: System state x(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' 5: Time-varying solar irradiance profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' worth noting that all the three algorithms can be classified as online methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' It has been widely perceived that online methods usually outperform offline counterparts in terms of conversion efficiency due to their inherent adaptiveness to changing environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' According to the curve-fitting based MPPT [42], the power and voltage (P-V ) characteristics can be modelled by P = φT(V )θ (71) where φ(V ) is the polynomial regressor [1, V, V 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' , V n]T and θ ∈ Rn+1 is the polynomial coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' To solve the maximum problem of (71), we need to estimate the unknown parameters θ and then maximise the power output by regulat- ing the voltage V according to V (k + 1) = V (k) + u(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' (72) We use solar panel A10Green Technology model number A10J-S72-175 for this simulation [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' To mimic the real operational environment of PV systems, a time-varying solar irradiance profile is stimulated as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' 5, and the temperature is initially set as 25°C and then jumps to 35°C at t = 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' It should be noted that the unknown environment parameter θ changes as the operational condition varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Al- though the proposed algorithm is theoretically analysed for static parameters identification, the use of constant learning rate ηi renders the adaptation algorithm in (15) with the capability of tracking drifting parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Simulation results using different algorithms (DCEE, HC and IC) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' 6, 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' To illustrate more de- tailed features of different algorithms, enlarged sub-figures are displayed for the time intervals t ∈ [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='1], and t ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The power losses, as displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' 9, are calculated by integrating the differences between the maximum power point and real power outputs stimulated using different algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Convergence speed, sensed signals, algorithm complexity and conversion efficiency are four commonly-used criteria to as- sess the characteristics of MPPT techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' According to the simulation results, we summarise and compare the features of different approaches in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Conversion efficiency directly influences the energy extracted from the PV systems, which is ratio between real generated energy and maximum energy (accumulated over the simulation time interval [0, 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' DCEE produces quite high efficiency (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='1%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Due to the use of perturbed signals in hill climbing method, there are very large voltage and current fluctuations in steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' This undesirable property not only causes low conversion efficiency but also leads to fast degradation in low level electronic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The oscillations are partially solved by incremental conductance method, which measures incremental current and voltage changes to predict the effect of voltage change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Different from HC, incremental inductance method is able to maintain at MPP without oscillations when there is no change in operational environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' From the simulation re- sults using HC and IC, there is a trade-off between transient convergence speed and steady-state oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The steady- state oscillation of IC is reduced at the cost of slow tracking performance, leading to larger power loss with a conversion efficiency 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' It is argued that DCEE as a balanced ap- proach is able to optimally trade-off between exploitation and exploration: when there is large uncertainty in estimated MPP, it will explore quickly to gain information to construct more accurate estimate of MPP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' and when there is less change in operational environment, it will maintain at the current belief of MPP without causing large oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' All three algorithms need to measure voltage and current: DCEE requires voltage and power (calculated by the product of current and voltage) to construct P-V curve in (71) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=', reward-state mapping), while HC and IC use incremental power to deicide the direction of voltage regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' As mature MPPT techniques, both HC and IC are simple to implement using dedicated hardware devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Since efficient ensemble approximation and gradient based control are developed in this new approach, DCEE is ready to be implemented in real PV platforms without incurring heavy computational load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' CONCLUSION In this paper, a general framework of dual control for ex- ploration and exploitation has been developed to solve a wide range of self-optimisation control problems in an uncertain environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' A real-time ensemble based estimation approach is proposed for efficient environment acquisition, which con- sequently provides a measure of knowledge uncertainty to the unknown environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' The proposed DCEE algorithm optimally balances between exploration and exploitation to 11 TABLE I: Features of different MPPT techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Methods Convergence speed Sensed variables Algorithm complexity Conversion efficiency 1 DCEE Fast Voltage and current Medium 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='1% 2 Hill climbing Fast Voltage and current Simple 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='3% 3 Incremental conductance Medium Voltage and current Simple 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content='2% Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' 6: Power profile using different algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' 7: Voltage profile using different algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' 8: Current profile using different algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' 9: Power losses using different algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' handle the intrinsic conflict between parameter identifiability and optimality tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Guaranteed convergence and perfor- mance are established in relation to the reward function and the noise characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' A numerical example and a classic application of MPPT are provided to validate the effectiveness and potential of DCEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' REFERENCES [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Zhang and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' Ordonez, “Numerical optimization-based extremum seeking control with application to ABS design,” IEEE Transactions on Automatic Control, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' 52, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfEi_Y/content/2301.11984v1.pdf'} +page_content=' 3, pp.' metadata={'source': 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Goddard, Clemson University +Chv´atal [1] defined the toughness of a graph G to be the minimum value of +|S|/k(G−S) where k(G−S) denotes the number of components of G−S and the +minimum is taken over all cut-sets S ⊆ V (G). It is immediate that the toughness +is at most half the connectivity. Matthews and Sumner [5] showed that there is +equality if the graph is claw-free. +For cubic graphs, Jackson and Katerinis [4] showed that being claw-free is +also necessary for the graph to have toughness 3 +2. In [2] we conjectured that the +analogous result holds for all r-regular graphs, and in [3] we expressed the belief +that the analogous result does not hold for all r, thus ensuring that we have to +be correct at least once. +We note here that it is the latter belief that is true. +The graph below is +4-regular and has claws and its toughness is 2. It is one of the two of smallest +order. +References +[1] V. Chv´atal. Tough graphs and Hamiltonian circuits. Discrete Math. 5 (1973), +215–28. +[2] W. Goddard and H.C. Swart. On the toughness of a graph. Quaestiones Math. +13 (1990), 217–232. +1 +arXiv:2301.13632v1 [math.CO] 27 Jan 2023 + +[3] W. Goddard. The toughness of cubic graphs. Graphs Combin. 12 (1996), 17– +22. +[4] B. Jackson and P. Katerinis. A characterization of 3 +2-tough cubic graphs. Ars +Combin. 38 (1994), 145–148. +[5] M.M. Matthews and D.P. Sumner. Hamiltonian results in K1,3-free graphs. +J. Graph Theory 8 (1984), 139–146. +2 + diff --git a/BNFRT4oBgHgl3EQfuzgt/content/tmp_files/load_file.txt b/BNFRT4oBgHgl3EQfuzgt/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2763467a1eca84b3d6fcb7abacc2854766f51dcb --- /dev/null +++ b/BNFRT4oBgHgl3EQfuzgt/content/tmp_files/load_file.txt @@ -0,0 +1,44 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf,len=43 +page_content='There are 2-tough 4-regular graphs with claws W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' Goddard, Clemson University Chv´atal [1] defined the toughness of a graph G to be the minimum value of |S|/k(G−S) where k(G−S) denotes the number of components of G−S and the minimum is taken over all cut-sets S ⊆ V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' It is immediate that the toughness is at most half the connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' Matthews and Sumner [5] showed that there is equality if the graph is claw-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' For cubic graphs, Jackson and Katerinis [4] showed that being claw-free is also necessary for the graph to have toughness 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' In [2] we conjectured that the analogous result holds for all r-regular graphs, and in [3] we expressed the belief that the analogous result does not hold for all r, thus ensuring that we have to be correct at least once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' We note here that it is the latter belief that is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' The graph below is 4-regular and has claws and its toughness is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' It is one of the two of smallest order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' References [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' Chv´atal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' Tough graphs and Hamiltonian circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' 5 (1973), 215–28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' [2] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' Goddard and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' Swart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' On the toughness of a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' Quaestiones Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' 13 (1990), 217–232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content='13632v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content='CO] 27 Jan 2023 [3] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' Goddard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' The toughness of cubic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' Graphs Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' 12 (1996), 17– 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' [4] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' Jackson and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' Katerinis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' A characterization of 3 2-tough cubic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' Ars Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' 38 (1994), 145–148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' Matthews and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' Sumner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' Hamiltonian results in K1,3-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' Graph Theory 8 (1984), 139–146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} +page_content=' 2' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFRT4oBgHgl3EQfuzgt/content/2301.13632v1.pdf'} diff --git 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In this paper, the main focus is +on an emergent ability in large vision models, +known as in-context learning, which allows +inference on unseen tasks by conditioning on +in-context examples (a.k.a. prompt) without +updating the model parameters. This concept has +been well-known in natural language processing +but has only been studied very recently for large +vision models. We for the first time provide a +comprehensive investigation on the impact of +in-context examples in computer vision, and find +that the performance is highly sensitive to the +choice of in-context examples. +To overcome +the problem, we propose a prompt retrieval +framework to automate the selection of in-context +examples. Specifically, we present (1) an unsuper- +vised prompt retrieval method based on nearest +example search using an off-the-shelf model, +and (2) a supervised prompt retrieval method, +which trains a neural network to choose examples +that +directly +maximize +in-context +learning +performance. The results demonstrate that our +methods can bring non-trivial improvements +to visual in-context learning in comparison to +the commonly-used random selection. +The +code and models are available at https: +//github.com/ZhangYuanhan-AI/ +visual_prompt_retrieval. +1. Introduction +In recent years, large-scale models have emerged in com- +puter vision: they have enormous parameter size and are +pre-trained on broad data to gain wide-ranging knowledge. +These models have demonstrated remarkable generaliza- +tion performance and have great potential for numerous +1S-Lab, Nanyang Technological University, Singapore. Corre- +spondence to: Ziwei Liu . +Preliminary work. Do not distribute. +downstream applications (Bommasani et al., 2021). How- +ever, due to the large model size and the potentially pro- +prietary data used for training, entities able to develop +large-scale models typically only provide users with APIs, +known as Model-as-a-Service (Maas). Representative exam- +ples include the prominent text-to-image generation models, +DALL·E (Ramesh et al., 2021) and Imagen (Saharia et al., +2022), and OpenAI’s powerful language models like GPT- +3/ChatGPT (Radford et al., 2021). As a result, users are +unable to apply full fine-tuning or some parameter-efficient +tuning techniques, such as prompt learning (Li & Liang, +2021; Lester et al., 2021; Zhou et al., 2022c;b; Zhang et al., +2022; Pan et al., 2022), for model adaptation, largely limit- +ing downstream performance. +In-context learning, which is a “hidden” capability origi- +nally found in large autoregressive language models (Rad- +ford et al., 2021), has recently been investigated for large +vision models (Bar et al., 2022), and more importantly, has +the potential to become the mainstream approach for MaaS +applications in the near future. Without the need to update +any parameter for previously unseen tasks, in-context learn- +ing simply prepends some domain-specific input-output +pairs, called in-context examples or prompt,1 to a test ex- +ample, which together guide the model to produce an ideal +result. For instance, in natural language processing one +could prepend a French-English sentence pair to a French +sentence, and the model would produce an English transla- +tion of the French sentence. In computer vision, Bar et al. +(2022) pre-trained a neural network to fill missing patches +in grid-like images, which allows the model to perform in- +context learning for unseen tasks like image segmentation +(see the grid images in Fig. 1(a) bottom). +In this work, we focus on visual in-context learning, a rel- +atively new concept with little existing research regarding +how to better apply it in practice. We for the first time +conduct a comprehensive investigation on the impact of in- +context examples for large vision models, and identify a +critical issue: downstream performance is highly sensitive +to the choice of in-context examples. This is evidenced by +the large variances observed for a variety of test examples +shown in Fig. 1(a) top. By visualizing the results in Fig. 1(a) +bottom, it seems to suggest that the closer the in-context +1These two terms are used interchangeably in this paper. +arXiv:2301.13670v1 [cs.CV] 31 Jan 2023 + +What Makes Good Examples for Visual In-Context Learning? +0 +20 +40 +60 +80 +100 +1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 +Foreground segmentation (mIoU) +Average +Standard deviation +Best prompt +Worst prompt +… +… +Example +Annotation +Query +Output +Database (train) +Query (test) +Large-scale +vision model +(b) Prompt retrieval for visual in-context learning +Score +function +Retrieved +image +(a)Visual in-context learning is sensitive to prompt selection +Query image index +Figure 1: (a) Different choices of in-context examples (outlined in green) often lead to significantly different results. Here +we show 30 random query images (x-axis) from Pascal-5i (Shaban et al., 2017) split 0, and measure the performance range +using 50 different in-context examples. (b) We propose a prompt retrieval framework aiming to automate the selection of +in-context examples. We provide two implementations of the idea: one is unsupervised while the other is supervised, both +outperforming random selection by a clear margin. +example to the query, the better the result. For example, the +best prompt image is closer to the query as they are similar +in object pose and background; on the other hand, the worst +prompt image has a drastically different style than the query +image, which might explain why the predicted mask focuses +on the wrong region, i.e., the white pillar instead of the cat. +Clearly, designing a proper prompt containing the optimal +in-context example(s) by hand would be extremely difficult. +To overcome the problem, we propose a prompt retrieval +framework where the core component is a score function, +which aims to give each source instance a score to indicate +the level of suitability for being included in the prompt. +Once the scoring process is done, we can simply pick one +or multiple examples with the highest score(s) to construct +a prompt. An overview of our framework is depicted in +Fig. 1(b). +We provide two implementations for the prompt retrieval +framework, both interpreting the score as the cosine distance +measuring similarity between a query and a source exam- +ple. The first is an unsupervised method based on nearest +example search using an off-the-shelf model. The second +is a supervised method, which learns a neural network to +choose examples that directly maximize in-context learning +performance. Since there is no ground-truth score to be +used as the supervisory signal, we resort to a contrastive +learning paradigm: source examples that result in better (or +worse) in-context learning performance should get closer +(or farther) to the query in feature space. +Our contributions and the main findings are summarized +as follows. (1) We present the first comprehensive study +concerning how to select good examples for the emerg- +ing visual in-context learning, and reveal a critical issue +that the choice of in-context examples has a huge impact +on performance. (2) From the technical perspective, we +present a prompt retrieval framework that can automate the +prompt selection process, and provide two simple implemen- +tations: an unsupervised method and a supervised method. +(3) By conducting extensive experiments on three visual +in-context learning tasks (which have not been seen dur- +ing pre-training), namely foreground segmentation, single +object detection and image colorization, we share valuable +insights with the community on how to find good visual +in-context examples, e.g., the supervised method performs +the best and often finds examples that are both semantically +close and spatially similar to a query. +2. Methods +2.1. Visual In-Context Learning +In-context learning is a new paradigm that originally +emerged from large autoregressive language models pre- +trained on broad data, such as GPT-3 (Brown et al., 2020). +Unlike traditional learning methods, in-context learning + +What Makes Good Examples for Visual In-Context Learning? +Feature +space +Learnable +feature +extractor +… +… +IoU:60.2% +IoU:30.1% +Query +🔥 +🧊 +Database (train) +Positive +Negative +Large-scale +vision model +Query (test) +high +low +🔥 +Figure 2: Overview of the supervised prompt retrieval method. The main idea is to compute the in-context learning result for +each source example, and pick those with the highest/lowest results to form a positive/negative set for contrastive learning. +does not require any parameter update and instead condi- +tions prediction on some in-context examples in the form +of input-output pairs. For example, in natural language pro- +cessing one might give a French-English sentence pair and +a test French sentence as input to the model, which then +produces the English version of the sentence. In computer +vision, such a paradigm has only been studied very recently. +For example, Bar et al. (2022) trained a neural network to +fill missing patches in grid-like images, which in turn allows +the model to perform in-context learning on unseen tasks. +Formally, given a dataset D = {(xn, yn)}N +n=1 containing +N image-label pairs (e.g., an image and its segmentation +mask), a query example xq, and a model gτ, in-context +learning can be formulated as: +yq = gτ(P, xq), +(1) +where P is called a prompt, which consists of K input- +output pairs, P = {xc1, yc1, ..., xcK, ycK} ⊂ D. In partic- +ular, the prompt P provides some context for guiding the +model to produce the ideal yq for xq without updating the +large model’s parameters τ. +Problem. The most common approach for designing the +prompt P in the vision domain is (within-class) random +selection proposed by Bar et al. (2022): one or multiple +image-label pairs (with the same label as the test example) +are randomly chosen from the training dataset. As illus- +trated in Fig. 1(a), the performance is highly sensitive to the +selection of in-context examples—the gap between the best +and worst prompt could reach over 70% mIoU. Below we +propose two automatic prompt selection methods to tackle +this problem. +2.2. Prompt Retrieval +Our goal is to automatically select the most suitable exam- +ple(s) from the training dataset for a query xq. To this end, +we propose a prompt retrieval framework in the following +form, +x∗ = arg max +xn∈D fθ(xn, xq), +(2) +where fθ is a function parameterized by θ, aiming to produce +a score for a pair of xn and xq. When K = 1, we choose +the optimal example pair as the prompt, P = {x∗, y∗}. +When K > 1, we rank the training examples by their scores +and choose the top-K example pairs. An overview of our +methods is provided in Fig. 1(b). +In this work, we implement fθ as a combination of a neural +network for feature extraction and the cosine distance func- +tion for measuring similarity between two feature vectors. +2.2.1. UNSUPERVISED PROMPT RETRIEVAL +Our first method is unsupervised prompt retrieval where +the key idea is to use an off-the-shelf feature extractor for +extracting image features so that we can compare the cosine +distance between the query xq and each training example +xn ∈ D. In this case, the parameters θ for the score function +fθ correspond to the off-the-shelf feature extractor, which +are kept fixed. +2.2.2. SUPERVISED PROMPT RETRIEVAL +The unsupervised method discussed above is not explicitly +optimized for in-context learning; instead, it relies on how +the feature extractor was pre-trained and the objective (func- +tion) used in pre-training may well not align with that of +in-context learning. We propose a second method based on + +AIR +CANADAWhat Makes Good Examples for Visual In-Context Learning? +supervised prompt retrieval where we assume the source +data contains labels. The goal is to directly optimize the +score function fθ such that the chosen in-context example(s) +can maximize the log-likelihood, +max +P +log p(yq|P, xq). +(3) +In this work, we present a simple implementation for the +supervised method, which simply turns the unsupervised +method into a supervised one by making the feature extrac- +tor learnable. In other words, we directly optimize Eq. 3 +with respect to the feature extractor. Below we explain in +detail how we train the feature extractor (see Fig. 2 for an +overview). +Data. Recall that we interpret the score fθ(·, ·) as the co- +sine distance between two images in feature space. We +would like to learn a space such that an image pair (xn, xq) +with high in-context learning performance is close to each +other, or far away from each other if the performance is +low. Since there is no label defining how close a distance +should be, we resort to contrastive learning for training the +feature extractor. The goal is then to find a positive and +a negative set for each training example xn ∈ D treated +as a query. Specifically, for each example xn we compute +the prediction ˆyn = gτ((xm, ym), xn) where gτ is the large +vision model defined in Sec. 2.1 and xm ∈ D but xm ̸= xn. +Since we have the ground truth yn for xn, we can measure +the performance by comparing the prediction ˆyn with the +ground truth yn. Then, for each xn we choose the top-5 +examples with the highest/lowest performance to form a +positive/negative set. +Training. Let zn denote the features of xn extracted by the +neural network we aim to optimize. At each iteraction, we +sample a mini-batch B from the training dataset. Then, for +each example in B, we sample one example from the top-5 +positive and negative sets, respectively. The contrastive loss +is computed as +ℓ = − 1 +|B| +� +xn∼B +log +ecos(zn,z+ +n ) +ecos(zn,z+ +n ) + +� +z− +n ∈N +ecos(zn,z− +n ) , (4) +where cos(·, ·) is the cosine distance function, z+ +n denotes +the feature representation of a positive example, and z− +n +denotes the feature representation of a negative example. It +is worth noting that for mini-batch training, the negative +set N contains a negative example of xn sampled from +the top-5 negative set and other examples within the same +mini-batch. +3. Experiments +In this section we conduct a comprehensive evaluation using +different prompt selection methods (Sec. 3.1) and compare +their robustness to distribution shifts (Sec. 3.2). We also +provide extensive quantitative and qualitative analyses in +Sec. 3.3 to help understand why our methods work and how +to better apply them in practice. Source code will be released +to the community for reproducing the full experiments. +Methods. All experiments are based on the image inpaint- +ing model pre-trained by Bar et al. (2022) on a dataset +consisting of academic figures.2 We mainly compare the fol- +lowing methods: (1) Random, the baseline method that ran- +domly samples in-context examples from the source training +dataset; (2) Unsupervised prompt retrieval (UnsupPR), our +first proposed method that uses off-the-shelf features for +nearest example search. The main experiments are based +on CLIP’s vision encoder (Radford et al., 2021), which was +pre-trained using multimodal contrastive learning; (3) Su- +pervised prompt retrieval (SupPR), our second proposed +method that fine-tunes CLIP’s vision encoder by directly +optimizing in-context learning performance on downstream +datasets. A variety of backbones are evaluated in Sec. 3.3. +Training details for the supervised model. The super- +vised model is trained for 200 epochs using SGD. The initial +learning rate is set to 0.005, decayed by the cosine annealing +rule. +3.1. Main Results +Setup. Following Bar et al. (2022), we evaluate our meth- +ods on three computer vision tasks, which have not been +seen during the training of the image inpainting model. We +provide the details about the datasets used for these tasks +as follows. (1) Foreground segmentation: We use Pascal- +5i (Shaban et al., 2017), which has four non-overlapping +splits each containing five categories. The results are aver- +aged over all splits. (2) Single object detection: The experi- +ments are done on Pascal VOC (Everingham et al., 2015). +(3) Colorization: We use ImageNet-2012 (Russakovsky +et al., 2015), where the original validation set containing +50,000 images is used as our test set. The training data used +to learn our supervised prompt retrieval model is created by +randomly sampling 50,000 images from ImageNet’s 1.2M +training set. For all experiments, in-context examples come +from the training set. +Results. Table 1 shows the results on the three benchmarks +covering foreground segmentation, single object detection, +and colorization. We summarize our findings as follows. +First, prompt retrieval clearly outperforms random selection. +In particular, the improvements of prompt retrieval over +random selection are significant in foreground segmentation +and single object detection: more than 6% on the former +2https://github.com/amirbar/visual_ +prompting + +What Makes Good Examples for Visual In-Context Learning? +Table 1: Main results. The two prompt retrieval methods outperform random selection, and the supervised method achieves +the best performance. +Seg. (mIoU) ↑ +Det. (mIoU) ↑ +Color. (mse) ↓ +Split-0 +Split-1 +Split-2 +Split-3 +Avg +Random +28.66 +30.21 +27.81 +23.55 +27.56 +25.45 +0.67 +UnsupPR +34.75 +35.92 +32.41 +31.16 +33.56 +26.84 +0.63 +SupPR +37.08 +38.43 +34.40 +32.32 +35.56 +28.22 +0.63 +Table 2: Results on distribution shifts (from Pascal to +MSCOCO). Despite being a learning-based approach, +SupPR shows stronger robustness than UnsupPR and Ran- +dom, which do not require any training. +Seg. (mIoU) ↑ +Split-0 +Split-1 +Split-2 +Split-3 +Avg +Random +12.17 +18.47 +20.55 +15.94 +16.78 +UnsupPR +12.67 +19.62 +21.33 +18.44 +18.02 +SupPR +13.62 +21.25 +24.46 +20.44 +19.95 +and 1% on the latter. However, the gains on colorization +are only marginal (0.63 vs. 0.67), suggesting that the image +inpainting model is probably weak at image colorization. +Second, the supervised prompt retrieval method performs +the best. This is not surprising as the supervised method +optimizes in-context learning performance concerning the +prompt selection module. In contrast, the unsupervised +method relies more on the off-the-shelf feature extractor. +Overall, the results well justify the design of the prompt +retrieval framework, which can serve as a strong baseline +for future research. +3.2. Experiments on Distribution Shifts +Setup. Distribution shifts are commonly seen in real-world +applications, and therefore AI models need to be robust to +distribution shifts (Zhou et al., 2022a). To test this ability in +visual in-context learning, we create a new protocol focusing +on foreground segmentation where the source dataset is +Pascal while the target dataset is MSCOCO (Lin et al., 2014). +Specifically, we follow the design of Pascal-5i and create +MSCOCO-5i, which also has four splits, each having the +same set of categories as in the corresponding split in Pascal- +5i. Note that such a shift mainly affects the supervised +prompt retrieval method that requires training but not the +unsupervised UnsupPR and Random. +Results. The results are shown in Table 2. First of all, +the unsupervised prompt retrieval method beats the random +selection method by a clear margin. By comparing the +two prompt retrieval methods, we find that the supervised +method again performs better than the unsupervised one +despite being a learning-based approach—this is an exciting +Table 3: Comparison between different backbones pre- +trained using different methods: multimodal contrastive +learning for CLIP, self-supervised learning for EVA, and +supervised learning for ViT. Overall, the performance is +insensitive to the choice of different backbones. +Seg. (mIoU) ↑ +Split-0 +Split-1 +Split-2 +Split-3 +Avg +UnsupPR +CLIP +34.75 +35.92 +32.41 +31.16 +33.56 +EVA +34.75 +36.09 +32.11 +31.61 +33.64 +ViT +35.10 +37.37 +32.05 +30.80 +33.83 +SupPR +CLIP +37.08 +38.43 +34.40 +32.32 +35.56 +EVA +36.11 +39.14 +34.31 +33.30 +35.71 +ViT +36.80 +39.70 +34.71 +33.25 +36.12 +finding as it means the supervised method does not have +the overfitting problem here. Nonetheless, we observe that +the gains achieved by the prompt retrieval methods here +are generally smaller than the gains achieved on the stan- +dard foreground segmentation benchmark: here SupPR is +only around 3% better on average than Random (19.95% +vs. 16.78%) while the improvement in Table 1 reaches 8% +(35.56% vs. 27.56%). One potential solution to reduce the +gap might be to improve the image inpainting model, which +is beyond the scope of this paper. +3.3. Further Analysis +What are good in-context examples? To answer this ques- +tion, we visualize the in-context examples found by Un- +supPR and SupPR in Fig. 3. We focus on foreground seg- +mentation and choose two categories from Pascal (person +and cow).3 In each grid, the first row corresponds to the re- +trieved in-context example (i.e., an input-output pair) while +the second row contains the query and model prediction. By +comparing the in-context examples picked by UnsupPR and +those picked by SupPR, we find the reason why SupPR per- +forms better than UnsupPR: the examples found by SupPR +are more similar to the queries in terms of semantics (e.g., +Fig. 3(e)), background (e.g., Fig. 3(a)), object pose (e.g., +Fig. 3(b), object appearance (e.g., Fig. 3(i)), viewpoint (e.g., +3The results of the remaining categories of Pascal and the +results on other tasks are provided in the supplementary. + +What Makes Good Examples for Visual In-Context Learning? +(g) +(h) +(i) +(j) +(k) +(l) +IoU: 37.85 +IoU: 47.48 +IoU: 42.36 +IoU: 69.46 +IoU: 26.47 +IoU: 27.78 +IoU: 59.34 +IoU: 46.74 +(a) +(b) +(c) +(d) +(e) +(f) +IoU: 49.12 +IoU: 23.21 +IoU: 66.93 +IoU: 61.25 +IoU: 29.34 +IoU: 63.38 +IoU: 8.45 +IoU: 36.67 +IoU: 86.44 +IoU: 86.64 +IoU: 92.32 +IoU: 80.14 +IoU: 63.14 +IoU: 79.22 +IoU: 57.48 +IoU: 49.87 +vvv +v v +vv +Figure 3: In-context examples retrieved by UnsupPR and SupPR. In each grid, the first row contains the prompt while the +second row contains the query and prediction. The in-context examples found by SupPR are more similar than those found +by UnsupPR to the queries in a numer of ways: semantics (e.g., (e)), background (e.g., (a)), object pose (e.g., (b), object +appearance (e.g., (i)), viewpoint (e.g., (k)), etc. More examples can be found in the supplementary. +Fig. 3(k)), and so on. We also observe similar patterns in +other categories/tasks (please refer to the supplementary). +Backbone. To understand if using a different backbone than +CLIP would make a big difference, we further evaluate our +prompt retrieval methods, UnsupPR and SupPR, on the fore- +ground segmentation benchmark using two other backbones: +EVA (Fang et al., 2022) pre-trained using self-supervised +learning (i.e., masked image modeling) and ViT (Dosovit- +skiy et al., 2020) pre-trained using supervised learning. The +results are reported in Table 3. Although these three back- +bones perform differently on image recognition under the +fine-tuning setting—EVA performed the best—the gap be- +tween them for both UnsupPR and SupPR is less than 1%. +Therefore, we can conclude that the backbone for visual +in-context learning does not matter much. +Size of retrieval set. Recall that in-context examples are +sampled from the training dataset, namely the retrieval set. +We are interested to know whether the size has any impact on +performance, especially for the supervised prompt retrieval +method. To this end, we build seven subsets for each split +in Pascal-5i, which cover a wide range of sizes (see the +x-axis in Fig. 4 left). The results are plotted in Fig. 4 left. +For random selection, the size does not matter at all. In +contrast, the two prompt retrieval methods clearly benefit +from a bigger size. But their performance plateaus when the +size reaches a certain level. It is worth noting that for the +supervised method, 20% of the total data is sufficient for +achieving a decent performance. + +What Makes Good Examples for Visual In-Context Learning? +27.58 +27.58 +27.66 +27.63 +27.83 +27.42 +27.56 +29.81 +32.01 +32.46 +32.85 +33.09 +33.50 +33.56 +31.30 +34.62 +35.42 +35.41 +35.55 +35.64 +35.56 +26 +27 +28 +29 +30 +31 +32 +33 +34 +35 +36 +0.01 +0.1 +0.2 +0.4 +0.6 +0.8 +1 +mIou +Size of retrieval set (% of full set) +Random +UnsupPR +SupPR +33.56 +35.56 +34.15 +33.70 +35.56 +34.03 +33.71 +35.57 +34.05 +33 +34 +35 +36 +UnsupPR +SupPR +AVG +mIOU +Similarity metric selection +Cosine +Euclidean +Manhattan +Figure 4: (Left) Impact of the size of retrieval set. (Right) Ablation study on distance metric used to compute the score +function in Eq. 2. It can be observed that different metrics perform similarly. +Table 4: Impact of the order of in-context examples. +Seg. (mIoU) ↑ +Split-0 +Split-1 +Split-2 +Split-3 +Avg +Random +17.93 ± 0.20 +25.48 ± 0.27 +21.34 ± 0.73 +21.12 ± 0.53 +21.46 ± 0.43 +UnsupPR +20.22 ± 0.31 +27.58 ± 0.40 +22.42 ± 0.38 +23.36 ± 0.42 +23.39 ± 0.37 +SupPR +20.74 ± 0.40 +28.19 ± 0.37 +23.09 ± 0.34 +24.22 ± 0.48 +24.06 ± 0.40 +Number of in-context examples. We follow Bar et al. +(2022) and create a large grid enough to fit 8 examples +at maximum (as shown in Fig. 5 right). By varying the +number of in-context examples from 1 to 7, we obtain a +set of results and plot them in Fig. 5 left. Clearly, more +in-context examples lead to better performance for all three +methods, including SupPR, UnsupPR, and Random. This +is probably because in-context examples can be viewed +as “training data”, and having more training data typically +benefits performance—in visual in-context learning, more +training data gives a more comprehensive “context.” We +show a few example cases in Fig. 5 right to explain this +observation. +Order of in-context examples. To understand if changing +the order of in-context examples makes a difference, we +fix the number of in-context examples to 3, evaluate all +possible combinations, and compute the mean and standard +deviation. As shown in Table 4, the standard deviation is +generally small, so the order is not a concern as long as +good examples are chosen. +Distance metric. We use the cosine distance by default to +compute the score function in Eq. 2. Here we evaluate other +design choices including Euclidean distance and Manhattan +distance. As shown in Fig. 4 right, the results are very +similar for different distance metrics. +4. Related Work +4.1. In-Context Learning +In-context learning is a novel paradigm that emerged in large +language models, such as GPT-3 (Brown et al., 2020). It al- +lows an autoregressive language model to perform inference +on unseen tasks by conditioning the input on some target- +specific input-output pairs serving as “context.” Such a pow- +erful paradigm allows users to customize a model’s output +according to their downstream datasets without changing +the internal model parameters, which are often inaccessi- +ble. Recent research in natural language processing has +shown that in-context learning can be applied to numerous +language tasks, such as machine translation (Garcia & Firat, +2022), sentiment analysis (Min et al., 2021), and question +answering (Press et al., 2022). +In computer vision, in-context learning is still a relatively +new concept. One of the earliest works tackling in-context +learning is Flamingo (Alayrac et al., 2022), a large visual +language model taking language as instruction and allowing +the processing of both images and videos. More relevant +to our work is a pure vision model developed by Bar et al. +(2022), which was pre-trained to fill missing patches in +images made of academic figures and infographics. Bar +et al. (2022) found that such an image inpainting model +can solve problems unseen during training, like foreground +segmentation and image colorization. +Our work follows Bar et al. (2022) but studies visual in- +context learning from a different dimension: how to find + +What Makes Good Examples for Visual In-Context Learning? +IoU: 20.98 +IoU: 15.68 +IoU: 32.50 +IoU:29.09 +(a) +Num. of examples = 1 +(b) +Num. of examples = 7 +18.56 +21.90 +22.87 +25.39 +21.22 +23.41 +24.39 +26.51 +21.90 +24.43 +25.87 +27.99 +18 +20 +22 +24 +26 +28 +1 +3 +5 +7 +mIoU +Number of in-context +examples +Random +UnsupPR +SupPR +(c) +IoU:34.19 +IoU:56.46 +Figure 5: (Left) Impact of the number of in-context examples. (Right) More in-context examples can lead to better +performance. The query in each grid is shown in the bottom right. +good visual in-context examples that benefit downstream +performance. +4.2. Prompt Retrieval in NLP +The natural language processing community has found that +the choice of in-context examples has a huge impact on per- +formance (Agrawal et al., 2022; Liu et al., 2021). Moreover, +the way how in-context examples, also called prompts, are +constructed can also affect performance, e.g., prompt length +and the order of in-context examples, as reported in the lit- +erature (Agrawal et al., 2022). These findings prompted the +community to study how to find good in-context examples +for large language models, which has inspired our research. +Liu et al. (2021) assumed that good in-context examples +should be semantically close to query sentences, based on +which they proposed to select nearest neighbors in the train- +ing set measured by a sentence encoder like RoBERTa (Liu +et al., 2019). Rubin et al. (2021) first used an unsupervised +method to retrieve some candidates, among which top ex- +amples were chosen using a supervised prompt retriever to +maximize downstream performance. +5. Discussion and Conclusion +Our research presents a timely study on an emergent abil- +ity termed in-context learning for large vision models. We +systematically investigate how the choice of in-context ex- +amples impacts downstream performance, exposing a crit- +ical issue that different in-context examples could lead to +drastically different results. We then propose an effective +prompt retrieval framework for visual in-context learning, +with two simple implementations provided: one based on +unsupervised learning and the other based on supervised +learning. Our methods obtain significant improvements over +random selection under various problem settings, showing +the potential of using prompt retrieval in vision applications +with a Model-as-a-Service (MaaS) business structure. +Our research also unveils some intriguing phenomena. For +instance, we show that a good in-context example should +be semantically similar to the query and closer in context, +e.g., viewpoint, background, and appearance. As such, state- +of-the-art vision models like CLIP would not be sufficient +because these models often emphasize semantics but not +other elements critical to finding good visual in-context +examples. A model that can better balance spatial and se- +mantic closedness in feature space would be more ideal for +visual in-context learning. We hope the insights presented in +this work could pave the way for developing more effective +prompt retrieval methods. +Our experiments show that our methods are not strong +enough to cope with distribution shifts. Though our meth- +ods outperform random selection under distribution shifts, +the gap is much smaller than that on a standard benchmark, +suggesting huge room for improvement. + +What Makes Good Examples for Visual In-Context Learning? +References +Agrawal, S., Zhou, C., Lewis, M., Zettlemoyer, L., and +Ghazvininejad, M. 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International Jour- +nal of Computer Vision (IJCV), 2022c. 1 + +What Makes Good Examples for Visual In-Context Learning? +A. Illustration of In-context Examples +In the supplementary material, we illustrate more in-context +learning results of foreground segmentation, single object +detection, and colorization tasks. +A.1. Foreground Segmentation +The main paper presents the in-context examples from the +person and cow categories. In the supplementary, as shown +in Fig. 6-11, we present examples from the remained 18 +categories in Pascal-5i. +A.2. Single Object Detection +As shown in Fig. 12-13, we illustrate the in-context ex- +amples from the single object detection task. By compar- +ing the in-context examples picked by UnsupPR and those +picked by SupPR, we find the examples found by SupPR +are more similar to the queries in terms of object pose (e.g., +Fig. 12(f)), viewpoint (e.g., Fig. 12(r). +A.3. Coloralization +As shown in Fig. 14-15, we illustrate the in-context exam- +ples from the colorization task. This task aims to map a +gray-scale image to a color image. By comparing the in- +context examples picked by UnsupPR and those picked by +SupPR, we find the ground truth images of examples found +by SupPR are more similar to that of the queries in terms of +image style, e.g. the background color (e.g., Fig. 14(g)(h)). + +What Makes Good Examples for Visual In-Context Learning? +(d) +IoU: 85.00 +IoU: 47.70 +IoU: 90.00 +(g) +(h) +(i) +(j) +(k) +(l) +(a) +(b) +(c) +(e) +(f) +IoU: 13.23 +IoU: 12.32 +IoU: 37.01 +IoU: 56.60 +IoU: 30.35 +IoU: 24.85 +IoU: 25.80 +IoU: 60.85 +IoU: 74.10 +IoU: 73.68 +IoU: 36.65 +IoU: 74.71 +IoU: 51.86 +IoU: 53.91 +IoU: 80.97 +IoU: 63.18 +IoU: 31.63 +IoU: 65.34 +IoU: 13.23 +IoU: 65,34 +IoU: 52.47 +IoU: 27.78 +IoU: 63.30 +IoU: 66.49 +IoU: 51.86 +IoU: 75.23 +IoU: 70.98 +IoU: 65.87 +IoU: 82.55 +IoU: 80.37 +IoU: 27.79 +IoU: 30.71 +IoU: 48.08 +IoU: 53.17 +(m) +(n) +(o) +(p) +(r) +(s) +Figure 6: In-context examples, which are from the foreground segmentation task, retrieved by UnsupPR and SupPR. These +grids show examples from the train, tv, and bus categories. + +What Makes Good Examples for Visual In-Context Learning? +(g) +(h) +(i) +(j) +(k) +(l) +(m) +(n) +(o) +(p) +(r) +(s) +IoU: 67.02 +IoU: 71.39 +IoU: 38.79 +IoU: 47.04 +IoU: 5.50 +• +IoU: 28.45 +IoU: 13.89 +IoU: 52.95 +(a) +(b) +(c) +(d) +(e) +(f) +IoU: 29.58 +IoU: 7.06 +IoU: 34.12 +IoU: 43.23 +IoU: 74.64 +IoU: 78.78 +IoU: 4.09 +IoU: 43.13 +IoU: 40.52 +IoU: 35.16 +IoU: 38.64 +IoU: 16.38 +IoU: 21.61 +IoU: 20.45 +IoU: 66.50 +IoU: 55.35 +IoU: 66.08 +IoU: 57.10 +IoU: 30.44 +IoU: 45.98 +IoU: 65.49 +IoU: 54.25 +IoU: 77.60 +IoU: 81.50 +IoU: 35.28 +IoU: 66.75 +IoU: 61.31 +IoU: 34.47 +Figure 7: In-context examples, which are from the foreground segmentation task, retrieved by UnsupPR and SupPR. These +grids show examples from the bottle, sheep, and bird categories. + +What Makes Good Examples for Visual In-Context Learning? +(g) +(h) +(i) +(j) +(k) +(l) +(m) +(n) +(o) +(p) +(r) +(s) +IoU: 6.695 +IoU: 21.72 +IoU: 8.06 +IoU: 26.55 +IoU: 5.50 +• +IoU: 6.96 +IoU: 13.89 +IoU: 25.93 +(a) +(b) +(c) +(d) +(e) +(f) +IoU: 5.24 +IoU: 11.12 +IoU: 1.96 +IoU: 61.63 +IoU: 43.06 +IoU: 52.62 +IoU: 62.24 +IoU: 58.01 +IoU: 28.07 +IoU: 56.08 +IoU: 17.95 +IoU: 57.14 +IoU: 38.60 +IoU: 7.50 +IoU: 34.96 +IoU: 71.44 +IoU: 29.17 +IoU: 70.87 +IoU: 74.21 +IoU: 14.59 +IoU: 41.31 +IoU: 64.74 +IoU: 52.98 +IoU: 63.29 +IoU: 68.04 +IoU: 72.49 +IoU: 8.52 +IoU: 15.43 +Figure 8: In-context examples, which are from the foreground segmentation task, retrieved by UnsupPR and SupPR. These +grids show examples from the boat, airplane, and bicycle categories. + +What Makes Good Examples for Visual In-Context Learning? +(d) +IoU: 59.41 +• +IoU: 67.96 +IoU: 76.11 +(g) +(h) +(i) +(j) +(k) +(l) +(m) +(n) +(o) +(p) +(r) +(s) +IoU: 0.00 +IoU: 40.91 +IoU: 10.53 +IoU: 23.59 +IoU: 5.49 +IoU: 28.56 +IoU: 28.38 +IoU: 42.44 +(a) +(b) +(c) +(e) +(f) +IoU: 23.46 +IoU: 0.25 +IoU: 33.00 +IoU: 28.87 +IoU: 34.29 +IoU: 62.26 +IoU: 9.00 +IoU: 50.00 +IoU: 0.00 +IoU: 0.00 +IoU: 63.77 +IoU: 63.17 +IoU: 26.03 +IoU: 5.00 +IoU: 84.00 +IoU: 75.39 +IoU: 79.47 +IoU: 51.00 +IoU: 48.62 +IoU: 51,92 +IoU: 72.58 +IoU: 92.00 +IoU: 57.00 +IoU: 26.25 +IoU: 36.67 +Figure 9: In-context examples, which are from the foreground segmentation task, retrieved by UnsupPR and SupPR. These +grids show examples from the car, cat, and chair categories. + +What Makes Good Examples for Visual In-Context Learning? +(d) +IoU: 22.37 +IoU: 65.46 +IoU: 65.50 +(g) +(h) +(i) +(j) +(k) +(l) +(m) +(n) +(o) +(p) +(r) +(s) +IoU: 0.00 +IoU: 54.00 +IoU: 0.00 +IoU: 68.04 +IoU: 25.92 +IoU: 21.99 +IoU: 70.16 +IoU: 38.34 +(a) +(b) +(c) +(e) +(f) +IoU: 48.93 +IoU: 73.70 +IoU: 41.00 +IoU: 54.02 +IoU: 41.22 +IoU: 60.00 +IoU: 42.66 +IoU: 26.76 +IoU: 39.71 +IoU: 67.09 +IoU: 17.91 +IoU: 22.38 +IoU: 59.27 +IoU: 71.61 +IoU: 85.94 +IoU: 49.44 +IoU: 71.77 +IoU: 68.91 +IoU: 65.63 +IoU: 60.82 +IoU: 70.87 +IoU: 72.97 +IoU: 71.77 +IoU: 51.60 +IoU: 77.78 +Figure 10: In-context examples, which are from the foreground segmentation task, retrieved by UnsupPR and SupPR. These +grids show examples from the dog, horse, and motorbike categories. + +14What Makes Good Examples for Visual In-Context Learning? +(d) +IoU: 0.00 +IoU: 63.09 +IoU: 6.45 +(g) +(h) +(i) +(j) +(k) +(l) +(m) +(n) +(o) +(p) +(r) +(s) +IoU: 31.88 +IoU: 45.71 +IoU: 34.58 +IoU: 0.00 +IoU: 60.62 +IoU: 27.09 +(a) +(b) +(c) +(e) +(f) +IoU: 0.74 +IoU: 8.34 +IoU: 23.25 +IoU: 3/00 +IoU: 43.73 +IoU: 27.15 +IoU: 29.16 +IoU: 2.44 +IoU: 56.38 +IoU: 0.00 +IoU: 2.23 +IoU: 28.82 +IoU: 62.40 +IoU: 41.35 +IoU: 93.05 +IoU: 39.02 +IoU: 45.35 +IoU: 65.69 +IoU: 18.00 +IoU: 46.55 +IoU: 33.91 +IoU: 35.21 +IoU: 33.75 +IoU: 47.07 +IoU: 41.56 +IoU: 7.08 +IoU: 38.61 +Figure 11: In-context examples, which are from the foreground segmentation task, retrieved by UnsupPR and SupPR. These +grids show examples from the table, plant, and sofa categories. + +What Makes Good Examples for Visual In-Context Learning? +(d) +IoU: 10.59 +IoU: 33.28 +IoU: 40.88 +(g) +(h) +(i) +(j) +(k) +(l) +(a) +(b) +(c) +(e) +(f) +IoU: 3.23 +IoU: 18.61 +IoU: 0.00 +IoU: 26.82 +IoU: 25.88 +IoU: 44.57 +IoU: 33.40 +IoU: 61.47 +IoU: 51.28 +IoU: 59.97 +IoU: 68.02 +IoU: 66.10 +IoU: 32.34 +IoU: 30.13 +IoU: 54.34 +IoU: 27.73 +IoU: 0.00 +IoU: 20.43 +IoU: 21.28 +IoU: 0.00 +IoU: 1.73 +IoU: 11.90 +IoU: 36.78 +IoU: 71.30 +IoU: 51.64 +IoU: 27.65 +IoU: 61.60 +IoU: 32.60 +IoU: 58.09 +IoU: 0.00 +IoU: 22.71 +IoU: 23.47 +IoU: 68.81 +(m) +(n) +(o) +(p) +(r) +(s) +Figure 12: In-context examples, which are from the single object detection task, retrieved by UnsupPR and SupPR. We find +the examples found by SupPR are more similar to the queries in terms of object pose (e.g., (f)), viewpoint (e.g., (r)) + +What Makes Good Examples for Visual In-Context Learning? +(d) +IoU: 31.79 +IoU: 32.98 +IoU: 63.17 +(g) +(h) +(i) +(j) +(k) +(l) +(a) +(b) +(c) +(e) +(f) +IoU: 3.23 +IoU: 36.21 +IoU: 13.14 +IoU: 15.34 +IoU: 6.89 +IoU: 39.50 +IoU: 14.15 +IoU: 73.74 +IoU: 29.59 +IoU: 67.14 +IoU: 67.26 +IoU: 69.75 +IoU: 32.34 +IoU: 53.40 +IoU: 71.97 +IoU: 52.23 +IoU: 26.33 +IoU: 20.43 +IoU: 0.00 +IoU: 20.72 +IoU: 41.23 +IoU: 4.51 +IoU: 8.05 +IoU: 71.30 +IoU: 29.61 +IoU: 42.64 +IoU: 7.53 +IoU: 7.26 +IoU: 45.97 +IoU: 0.00 +IoU: 45.88 +IoU: 24.76 +IoU: 58.6 +(m) +(n) +(o) +(p) +(r) +(s) +c +Figure 13: In-context examples, which are from the single object detection task, retrieved by UnsupPR and SupPR. We find +the examples found by SupPR are more similar to the queries in terms of object pose (e.g., (l)), viewpoint (e.g., (m)) + +BAGGAGEWhat Makes Good Examples for Visual In-Context Learning? +mse : 0.79 +mse : 0.88 +mse : 1.13 +mse: 0.91 +mse : 0.85 +mse : 0.63 +(d) +mse : 0.49 +(a) +(b) +(c) +(e) +(f) +mse : 0.53 +mse : 0.67 +mse : 0.52 +mse : 0.78 +mse : 0.51 +mse : 1.15 +mse : 0.68 +mse : 0.76 +mse : 2.16 +mse : 0.99 +mse : 0.38 +(g) +(h) +(i) +(j) +(k) +(l) +mse : 0.88 +mse : 0.73 +mse : 0.23 +mse : 0.17 +mse : 0.40 +mse : 0.48 +Figure 14: In-context examples, which are from the colorization task, retrieved by UnsupPR and SupPR. We also show the +ground truth of the query image. The query image is the gray-scale version of its ground truth. The ground truth images +of the in-context examples found by SupPR are more similar than those found by UnsupPR to the ground truth images of +queries in terms of image style, e.g. the background color (g). + +What Makes Good Examples for Visual In-Context Learning? +mse : 0.66 +mse : 1.15 +mse : 1.51 +mse: 1.01 +mse : 0.78 +mse : 0.79 +(d) +mse : 0.44 +(a) +(b) +(c) +(e) +(f) +mse : 0.80 +mse : 0.64 +mse : 0.80 +mse : 0.55 +mse : 0.51 +mse : 3.48 +mse : 1.19 +mse : 0.79 +mse : 0.88 +mse : 0.80 +mse : 0.99 +(g) +(h) +(i) +(j) +(k) +(l) +mse : 0.36 +mse : 0.52 +mse : 0.85 +mse : 0.73 +mse : 0.49 +mse : 0.34 +Figure 15: In-context examples, which are from the colorization task, retrieved by UnsupPR and SupPR. We also show the +ground truth of the query image. The query image is the gray-scale version of its ground truth. The ground truth images +of the in-context examples found by SupPR are more similar than those found by UnsupPR to the ground truth images of +queries in terms of image style, e.g. the background color (h). + +odidasacnosadidas +adidas +adidos \ No newline at end of file diff --git a/OdFRT4oBgHgl3EQf4zhC/content/tmp_files/load_file.txt b/OdFRT4oBgHgl3EQf4zhC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9c51cf97f1ce1cf58d05f106a604ece07eb73e72 --- /dev/null +++ b/OdFRT4oBgHgl3EQf4zhC/content/tmp_files/load_file.txt @@ -0,0 +1,1276 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQf4zhC/content/2301.13670v1.pdf,len=1275 +page_content='What Makes Good Examples for Visual In-Context Learning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQf4zhC/content/2301.13670v1.pdf'} +page_content=' Yuanhan Zhang 1 Kaiyang Zhou 1 Ziwei Liu 1 Abstract Large-scale models trained on broad data have recently become the mainstream architecture in computer vision due to their strong generalization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQf4zhC/content/2301.13670v1.pdf'} +page_content=' In this paper, the main focus is on an emergent ability in large vision models, known as in-context learning, which allows inference on unseen tasks by conditioning on in-context examples (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQf4zhC/content/2301.13670v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQf4zhC/content/2301.13670v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQf4zhC/content/2301.13670v1.pdf'} +page_content=' prompt) without updating the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQf4zhC/content/2301.13670v1.pdf'} +page_content=' This concept has been well-known in natural language processing but has only been studied very recently for large vision models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQf4zhC/content/2301.13670v1.pdf'} +page_content=' We for the first time provide a comprehensive investigation on the impact of in-context examples in computer vision, and find that the performance is highly sensitive to the choice of in-context examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQf4zhC/content/2301.13670v1.pdf'} +page_content=' To overcome the problem, we propose a prompt retrieval framework to automate the selection of in-context examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQf4zhC/content/2301.13670v1.pdf'} +page_content=' Specifically, we present (1) an unsuper- vised prompt retrieval method based on nearest example search using an off-the-shelf model, and (2) a supervised prompt retrieval method, which trains a neural network to choose examples that directly maximize in-context learning performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQf4zhC/content/2301.13670v1.pdf'} +page_content=' The results demonstrate that our methods can bring non-trivial improvements to visual in-context learning in comparison to the commonly-used random selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQf4zhC/content/2301.13670v1.pdf'} +page_content=' The code and models are available at https: //github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQf4zhC/content/2301.13670v1.pdf'} +page_content='com/ZhangYuanhan-AI/ visual_prompt_retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQf4zhC/content/2301.13670v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQf4zhC/content/2301.13670v1.pdf'} +page_content=' Introduction In recent years, large-scale models have emerged in com- puter vision: they have enormous parameter size and are pre-trained on broad data to gain wide-ranging knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQf4zhC/content/2301.13670v1.pdf'} +page_content=' These models have demonstrated remarkable generaliza- tion performance and have great potential for numerous 1S-Lab, Nanyang Technological University, Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFRT4oBgHgl3EQf4zhC/content/2301.13670v1.pdf'} +page_content=' Corre- spondence to: Ziwei Liu . +Preprint. +2021; Lee et al., 2019; 2021), playing board games (Silver +et al., 2016; 2017), and strategy games (Vinyals et al., 2019; +Wurman et al., 2022). Yet, the black-box nature of neural +network-based policies makes it difficult for the DRL-based +systems to be interpreted and therefore trusted by human +users (Lipton, 2016; Puiutta & Veith, 2020). Moreover, poli- +cies learned by DRL methods tend to overfit and often fail +to generalize (Zhang et al., 2018; Cobbe et al., 2019; Sun +et al., 2020; Liu et al., 2022). +To address the abovementioned issues of DRL, program- +matic RL methods (Bastani et al., 2018; Inala et al., 2020; +Landajuela et al., 2021; Verma et al., 2018) explore various +of more structured representation of policies. In particu- +lar, Trivedi et al. (2021) present a framework, Learning +Embeddings for lAtent Program Synthesis (LEAPS), that +is designed to produce more interpretable and generaliz- +able policies. Specifically, it aims to produce program poli- +cies structured in a given domain specific language (DSL), +which can be executed to yield desired behaviors. To this +end, LEAPS first learns a program embedding space to +continuously parameterize diverse programs from a pre- +generated program dataset, and then searches for a task- +solving program in the learned program embedding space +when given a task described by a Markov decision process +(MDP). The program policies produced by LEAPS are not +only human-readable but also achieve competitive perfor- +mance and demonstrate superior generalization ability. +Despite its encouraging results, LEAPS has two fundamen- +tal limitations. Limited program distribution: the program +policies that LEAPS can produce are limited by the distribu- +tion of the pre-generated program dataset used for learning +the program embedding space. This is because LEAPS is de- +signed to search for a task-solving program from the learned +embedding space, which inherently assumes that such a +program is within the distribution of the program dataset. +Such design makes it difficult for LEAPS to synthesize pro- +grams that are out-of-distributionally long or complex. Poor +credit assignment: during the search for the task-solving +program embedding, LEAPS evaluates each candidate pro- +gram solely based on cumulative discounted return of the +program execution trace. Such design fails to accurately +attribute rewards obtained during the execution trajectories +arXiv:2301.12950v1 [cs.LG] 30 Jan 2023 + +Hierarchical Programmatic Reinforcement Learning +to corresponding parts in synthesized programs or penalize +program parts that induce incorrect behaviors. +This work aims to address the issues of limited program +distribution and poor credit assignment. To this end, we +propose a hierarchical programmatic reinforcement learning +(HPRL) framework. Instead of searching for a program +from a learned program embedding space, we propose to +learn a meta-policy, whose action space is the learned pro- +gram embedding space, to produce a series of programs +(i.e., predict a sequence of actions) to yield a composed +task-solving program. By re-formulating synthesizing a +program as predicting a sequence of programs, HPRL can +produce out-of-distributionally long or complex programs. +Furthermore, rewards obtained from the environment by +executing each program from the composed program can +be accurately attributed to the program, allowing for more +efficient learning. +To evaluate our proposed method, we adopt the Karel do- +main (Pattis, 1981), which features an agent that can nav- +igate a grid world and interact with objects. Our method +outperforms all the baselines by large margins on a prob- +lem set proposed in (Trivedi et al., 2021). To investigate +the limitation of our method, we design a more challeng- +ing problem set on which our method consistently achieves +better performance compared to LEAPS. Moreover, we in- +spect LEAPS’ issues of limited program distribution and +poor credit assignment with two experiments and demon- +strate that our proposed method addresses these issues. We +present a series of ablation studies to justify our design +choices, including the reinforcement learning algorithms +used to learn the meta-policy, and the dimensionality of the +program embedding space. +2 +Related Work +Program Synthesis. Program synthesis methods concern +automatically synthesize programs that can transform some +inputs to desired outputs. Encouraging results have been +achieved in a variety of domains, including string transfor- +mation (Devlin et al., 2017; Hong et al., 2021), array/tensor +transformation (Balog et al., 2017; Ellis et al., 2020), com- +puter commands (Lin et al., 2018; Chen et al., 2021; Li et al., +2022), graphics and 3D shape programs (Wu et al., 2017; +Liu et al., 2019; Tian et al., 2019), and describing behav- +iors of agents (Bunel et al., 2018; Sun et al., 2018; Shin +et al., 2018; Chen et al., 2019; Liao et al., 2019; Silver et al., +2020). Most existing program synthesis methods consider +task specifications such as input/output pairs, demonstra- +tions, or natural language descriptions. In contrast, we aim +to synthesize programs as policies that can be executed +to induce behaviors which maximize rewards defined by +reinforcement learning tasks. +Program ρ := DEF run m( s m) +Repetition n := Number of repetitions +Perception h := frontIsClear | leftIsClear | rightIsClear | +markerPresent | noMarkerPresent +Condition b := perception h | not perception h +Action a := move | turnLeft | turnRight | +putMarker | pickMarker +Statement s := while c( b c) w( s w) | s1; s2 | a | +repeat R=n r( s r) | if c( b c) i( s i) | +ifelse c( b c) i( s1 i) else e( s2 e) +Figure 1. The domain-specific language (DSL) for the Karel do- +main, features an agent that can navigate through a grid world and +interact with objects. +Programmatic Reinforcement Learning. Programmatic +reinforcement learning methods (Choi & Langley, 2005; +Winner & Veloso, 2003; Sun et al., 2020) explore various +programmatic and more structured representations of poli- +cies, including decision trees (Bastani et al., 2018), state +machines (Inala et al., 2020), symbolic expressions (Lan- +dajuela et al., 2021), and programs drawn from a domain- +specific language (Silver et al., 2020; Verma et al., 2018; +2019). Our work builds upon (Trivedi et al., 2021), whose +goal is to produce program policies from rewards. We aim to +address the fundamental limitations of this work by learning +to compose programs to yield more expressive programs. +Hierarchical Reinforcement Learning. Hierarchical rein- +forcement learning (HRL) (Sutton et al., 1999; Barto & +Mahadevan, 2003; Vezhnevets et al., 2017; Bacon et al., +2017) aims to learn to operate on different levels of tem- +poral abstraction, allowing for learning or exploring more +efficiently in sparse-reward environments. In this work, +instead of operating on pre-defined or learned temporal +abstraction, we are interested in learning with a level of ab- +straction defined by a learned program embedding space to +hierarchically compose programs. One can view a learned +program embedding space as continuously parameterized +options or low-level policies. +3 +Problem Formulation +Our goal is to develop a method that can synthesize a +domain-specific, task-solving program which can be ex- +ecuted to interact with an environment and maximize a +discounted return defined by a Markov Decision Process. +Domain Specific Language. In this work, we adapt the +domain specific language (DSL) for the Karel domain used +in (Bunel et al., 2018; Chen et al., 2019; Trivedi et al., 2021), +shown in Figure 1. This DSL is designed to describe the +behaviors of the Karel agent, consisting of control flows, + +Hierarchical Programmatic Reinforcement Learning +agent’s perceptions, and agent’s actions. Control flows such +as if, else, and while are allowed for describing di- +verging or repetitive behaviors. Furthermore, Boolean and +logical operators such as and, or, and not can be included +to express more sophisticated conditions. Perceptions such +as frontIsClear and markerPresent are defined +based on situations in an environment which can be per- +ceived by an agent. On the other hand, actions such as +move, turnRight, and putMarker, describe the prim- +itive behaviors that an agent can perform in an environment. +A program policy considered in our work is structured in +this DSL and can be executed to produce a sequence of +actions based on perceptions. +Markov Decision Process (MDP). The tasks considered in +this work are defined by finite-horizon discounted MDPs. +The performance of a policy with its rollout (a sequence of +states and actions {(s0, a0), ..., (st, at)}) is evaluated based +on a discounted return �T +t=0 γtrt, where rt = R(st, at) +indicates the reward function and T is the horizon of the +episode. We aim to develop a method that can produce a pro- +gram representing a policy that can be executed to maximize +the discounted return, i.e., maxρ Ea∼EXEC(ρ)[�T +t=0 γtrt], +where EXEC(·) returns the actions induced by executing +the program policy ρ in the environment. This objective is a +special case of the standard RL objective where a policy is +represented as a program in a DSL and the policy rollout is +obtained by executing the program. +4 +Approach +Our goal is to design a framework that can synthesize task- +solving programs based on the rewards obtained from MDPs. +We adapt the idea of learning a program embedding space +to continuously parameterized a diverse set of programs +proposed in LEAPS (Trivedi et al., 2021). Then, instead of +searching for a task-solving program in the learned program +embedding space, our key insight is to learn a meta-policy +that can hierarchically compose programs to form a more +expressive task-solving program. Our proposed framework, +dubbed Hierarchical Programmatic Reinforcement Learning +(HPRL), is capable of producing out-of-distributionally long +and complex programs. Moreover, HPRL can make delicate +adjustments to synthesized programs according to rewards +obtained from the environment. +Section 4.1 presents how LEAPS learns a program embed- +ding space to continuously parameterize a set of randomly +generated programs and describes our proposed procedure to +produce a dataset containing more diverse programs. Then, +to reduce the dimension of the learned program embedding +for more efficient meta-policy learning, Section 4.2 intro- +duces how we compress the embedding space. Finally, in +Section 4.3, we describe our method for learning a meta- +policy, whose action space is the learned program embed- +ding space, to hierarchically compose programs and yield a +task-solving program. An overview of our proposed frame- +work is illustrated in Figure 2 and the algorithm is detailed +in Algorithm 1. +4.1 +Learning a Program Embedding Space +We aim to learn a program embedding space that continu- +ously parameterizes a diverse set of programs. Moreover, a +desired program embedding space should be behaviorally +smooth, i.e., programs that induce similar execution traces +should be embedded closely to each other and programs +with diverging behaviors should be far from each other in +the embedding space. +To this end, +we adapt the technique proposed in +LEAPS (Trivedi et al., 2021), which trains an encoder- +decoder neural network architecture on a pre-generated +program dataset. Specifically, a recurrent neural network +program encoder qφ learns to encode a program ρ (i.e., +sequences of program tokens) into a program embedding +space, yield a program embedding v; a recurrent neural +network program decoder pθ learns to decode a program +embedding v to produce reconstructed programs ˆρ. The +program encoder and the program decoder are trained to +optimize the β-VAE (Higgins et al., 2016) objective: +LP +θ,φ(ρ) = −Ev∼qφ(v|ρ)[log pθ(ρ|v)] ++βDKL(qφ(v|ρ)∥pθ(v)), +(1) +where β balances the reconstruction loss and the representa- +tion capacity of the embedding space (i.e., the latent bottle- +neck). +To encourage behavioral smoothness, Trivedi et al. (2021) +propose two additional objectives. The program behavior +reconstruction loss minimizes the difference between the +execution traces of the given program EXEC(ρ) and the +execution traces of the reconstructed program EXEC(ˆρ). +On the other hand, the latent behavior reconstruction loss +brings closer the execution traces of the given program +EXEC(ρ) and the execution traces produced by feeding the +program embedding v to a learned neural program executor +π(a|v, s): +LL +π(ρ, π) = −E[ +H +� +t=1 +|A| +� +i=1 +1{EXECi(ˆρ) == EXECi(ρ)} +log π(ai|v, st)], +(2) +where H denotes the horizon of EXEC(ρ) and |A| denotes +the cardinality of the action space. +We empirically found that optimizing the program behavior +reconstruction loss does not yield a significant performance +gain. Yet, due to the non-differentiability nature of program + +Hierarchical Programmatic Reinforcement Learning +(a) Learning a Program Embedding Space +(b) Learning a Meta-Policy to Compose Programs +i-th Predicted +Latent Program +Execute ρi +pθ +gψ +zi +Environment +si+1 +ri+1 +πmeta +si +t +ai +t +[si +1,   . . . ,  si +Ti] +[ri +1,   . . . ,  ri +Ti] +[-1] +⋅ +Σ +def run(): +while(markPresent()): +PickMarker() +turnRight() +move() +ρ1 +def run(): +if frontIsClear(): +move() +else: +turnLeft() +ρi−2 +def run(): +if frontIsClear(): +move() +else: +turnLeft() +ρi−1 +def run(): +if markerPresent(): +pickMarker() +else: +move() +i-th Predicted Program ρi +Composed Program +𝒫 = ⟨ρ1,   . . . ,  ρi−2,  ρi−1,  ρi⟩ +Latent +Program +Program ρ +def run(): +if markerPresent(): +pickMarker() +else: +move() +def run(): +if markerPresent(): +pickMarker() +else: +move() +LP +Reconstructed +Program ̂ρ +Execute +LL +qϕ +pθ +fω +gψ +z +π(a|s, z) +Environment +a1,  a2,   . . . ,  at +̂a1,   ̂a2,   . . . ,   ̂at +Learning objective +Learnable mapping +Frozen mapping +List operator +Compose +Figure 2. Hierarchical Programmatic Reinforcement Learning. (a) Learning a Program Embedding Space: a continuously param- +eterized latent program space can be learned using the program encoder qφ, decoder pθ, and a neural executor policy π by optimizing the +two reconstruction objectives: LP and LL. To reduce the dimensionality of the program embedding space for facilitate task learning, we +employ a compression encoder fω and a compression decoder gψ. (b) Learning a Meta-Policy to Compose Programs: given a task +described by an MDP, we propose to train a meta-policy πmeta to compose a sequence of programs, and yield a task-solving program. +Specifically, at each macro time step i, the meta-policy πmeta predicts a latent program embedding zi, which can be decoded to the +corresponding program ρi = pθ(gψ(zi)). We then execute the program ρi in the environment, which returns the cumulative reward ri+1 +and the next state si+1 to the meta policy. The meta-policy can synthesize next program ρi+1 based on si+1 to synthesize the task-solving +program P = ⟨ρ1, .., ρi−2, ρi−1, ρi⟩, until termination. +execution, optimizing this loss via REINFORCE (Williams, +1992) is unstable. Moreover, performing on-the-fly program +execution during training significantly slows down the learn- +ing process. Therefore, we exclude the program behavior +reconstruction loss, yielding our final objective for learning +a program embedding space as a combination of the β-VAE +objective LP +θ,φ and the latent behavior reconstruction loss +LL +π: +min +θ,φ,π LP +θ,φ(ρ) + λLL +π(ρ, π), +(3) +where λ determines the relative importance of these losses. +4.2 +Compressing the Learned Program Embedding +Space +The previous section describes a method for constructing a +program embedding space that continuously parameterizes +programs. Next, given a task defined by an MDP, we aim +to learn a meta-policy that predicts a sequence of program +embeddings as actions to compose a task-solving program. +Hence, a low-dimensional program embedding space (i.e., +a smaller action space) is ideal for efficiently learning such +a meta-policy. Yet, to embed a large number of programs +with diverse behaviors, a learned program embedding space +needs to be extremely high-dimensional. +Therefore, our goal is to bridge the gap between a high- +dimensional program embedding space with sufficient rep- +resentation capacity and a desired low-dimensional action +space for learning a meta-policy. To this end, we propose +to learn to compress the program embedding space with +an encoder-decoder architecture. Specifically, we employ +a compression encoder fω that takes the output of the pro- +gram encoder qφ as input and compresses it into a lower- +dimensional program embedding z; also, we employ a com- +pression decoder gψ that takes a program embedding as +input and decompresses it to produce a reconstructed higher- +dimensional program embedding ˆv, which is then fed to the +program decoder pθ to produce a reconstructed program ˆρ. +With this modification, the β-VAE objective and the latent +behavior reconstruction loss can be rewritten as: +LP +θ,φ,ω,ψ(ρ) = −Ez∼fω(z|qφ(ρ))[log pθ(ρ|(gψ(z)))] ++βDKL(fω(qφ(z|ρ))∥pθ(gψ(z))), +(4) +and +LL +π(ρ, π) = −E[ +H +� +t=1 +|A| +� +i=1 +1{EXECi(ˆρ) == EXECi(ρ)} +log π(ai|z, st)]. +(5) +We train the program encoder qφ, the compression encoder + +Hierarchical Programmatic Reinforcement Learning +fω, the compression decoder gψ, the program decoder pθ, +and the neural execution policy π in an end-to-end manner. +We discuss how the dimension of the program embedding +space affects the quality of program reconstruction and the +performance of synthesized programs in Section 5.4.2. +4.3 +Learning a Meta-Policy to Compose the +Task-Solving Program +Once a expressive, smooth, yet compact program embed- +ding space is learned, given a task described by an MDP, +we propose to train a meta-policy πmeta to compose a task- +solving program. Specifically, the learned program em- +bedding space is used as a continuous action space for the +meta-policy πmeta, bounded within the range of [-1.0, 1.0] +for each dimension of the program embedding. We for- +mulate the task of composing programs as a finite-horizon +MDP whose horizon is |H|. At each time step i, the meta- +policy πmeta takes an input state si, and predicts one latent +program embedding zi as action, which can be decoded to +its corresponding program ρi using the learned compres- +sion decoder and program decoder pθ(gψ(zi)). Then, the +program ρi is executed with EXEC(·) to interact with the +environment for a period from 1 to T i, yielding the cumu- +lative reward ri+1 = �T i +t=1 ri +t and the next state si+1 = +[si +1, si +2, ..., si +Ti][−1] after the program execution. The opera- +tor ·[−1] returns the last object in the sequence, and we take +the last state of the program execution as the next macro in- +put state, i.e, si+1 = [si +1, si +2, ..., si +Ti][−1] = si +Ti. Note that +the time steps i considered here are macro time steps, each +involves a series of state transitions and returns a sequence +of rewards. The environment will return the next state si+1 +and cumulative reward ri+1 to the agent to predict the next +latent program embedding zi+1. The program composing +process terminates after repeating |H| steps. +The synthesized task-solving program P is obtained by se- +quentially compose the generated program ⟨ρi|i = 1...|H|⟩, +where ⟨·⟩ denotes an operator that concatenates programs +in order to yield a composed program. Hence, the learning +objective of the meta-policy πmeta is to maximize the total +cumulative return Jπmeta: +Jπmeta = EP∼πMETA[ +|H| +� +i=1 +γi−1Ea∼EXEC(ρi)[ri+1]. +(6) +where γ is the discount factor for macro time steps MDP +and a is the primitive action triggered by EXEC(ρi). +While this work formulates the program synthesis task as a +finite-horizon MDP where a fixed number of programs are +composed, we can instead learn a termination function that +decides when to finish the program composition process, +which is left to future work. +(a) DOORKEY +(b) ONESTROKE +(c) SEEDER +(d) SNAKE +Figure 3. KAREL-HARD Problem Set: The four tasks require +an agent to acquire a set of diverse, goal-oriented, and program- +matic behaviors. This is strictly more challenging compared to +the KAREL problem set proposed in (Trivedi et al., 2021). More +details can be found in Section B. +5 +Experiments +We design and conduct experiments to compare our pro- +posed framework (HPRL) to its variants and baselines. +5.1 +Karel domain +For the experiments and ablation studies, we adopt the +Karel domain (Pattis, 1981), which is widely used in neural +program synthesis and programmatic reinforcement learn- +ing (Bunel et al., 2018; Shin et al., 2018; Sun et al., 2018; +Chen et al., 2019; Trivedi et al., 2021). The Karel agent +in a gridworld can navigate and interact with objects (i.e., +markers). The action and perception is detailed in Figure 1. +To evaluate the proposed framework and the baselines, we +consider two problem sets. First, we use the KAREL prob- +lem set proposed in (Trivedi et al., 2021), which consists of +six tasks. Then, we propose a more challenging set of tasks, +KAREL-HARD problem set (shown in Figure 3), which con- +sists of four tasks. In most tasks, initial configurations such +as agent and goal locations, wall and marker placement, are +randomly sampled upon every episode reset. +KAREL Problem Set. +The KAREL problem set intro- +duced in (Trivedi et al., 2021) consists of six tasks: STAIR- +CLIMBER, FOURCORNER, TOPOFF, MAZE, CLEAN- +HOUSE and HARVESTER. Solving these tasks requires +the following ability. Repetitive Behaviors: to conduct +the same behavior for several times, i.e., placing markers +on all corners (FOURCORNER) or move along the wall +(STAIRCLIMBER). +Exploration: to navigate the agent +through complex patterns (MAZE) or multiple chambers + +Hierarchical Programmatic Reinforcement Learning +(CLEANHOUSE). Complexity: to perform specific actions, +i.e., put markers on marked grid (TOPOFF) or pick markers +on markerd grid (HARVESTER). For further description +about the KAREL problem set, please refer to Section B.1. +KAREL-HARD Problem Set. We design a more challeng- +ing set of tasks, the KAREL-HARD problem set. The ability +required to solve the tasks in this problem set can be cate- +gorized as follows: Two-stage exploration: to explore the +environment under different conditions, i.e., pick up the +marker in one chamber to unlock the door, and put the +marker in the next chamber (DOORKEY). Additional Con- +straints: to perform specific actions under restrictions, i.e., +traverse the environment without revisiting the same posi- +tion (ONESTROKE), place exactly one marker on all grids +(SEEDER), and traverse the environment without hitting a +growing obstacle (SNAKE). More details about the KAREL- +HARD problem set can be found in Section B.8. +5.2 +Experimental Settings +Section 5.2.1 introduces the procedure for generating the +program dataset used for learning a program embedding +pace. The implementation of the proposed framework is +described in Section 5.2.2. +5.2.1 +Karel DSL Program Dataset Generation with +Our Improved Generation Procedure +The Karel program dataset used in this work includes one +million programs. All the programs are generated based on +syntax rules of the Karel DSL with a maximum length of 40 +program tokens. While Trivedi et al. (2021) randomly sam- +ple to generate program sequences, we propose an improved +program generation procedure as follows. We filter out +counteracting programs (e.g. termination state equals initial +state after program execution), repetitive programs (e.g. , +programs with long common sub-sequences) and programs +with canceling action sequences (e.g., turnLeft followed +by turnRight). These rules significantly improve the +diversity and expressiveness of the generated programs and +induce a more diverse and complex latent program space. +More details can be found in Section D. +5.2.2 +Implementation +Encoders & Decoders. We use GRU (Cho et al., 2014) +layer to implement both the program encoder qφ and the +program decoder pθ mentioned in Section 4.1 with a hidden +dimension of 256. The last hidden state of the encoder qφ +is taken as the uncompressed program embedding v. This +program embedding v can be further compressed to a 64- +dimensional program embedding z using the compression +encoder fω and compression decoder gψ constructed by the +fully-connected neural network as described in Section 4.2. +Neural Program Executor. The neural program executor +π is implemented as a recurrent conditional policy π(·|z, s) +using GRU layers, which takes the abstract state and the +program embedding z at each time step as input and predicts +the execution trace. +Meta-Policy. +To implement the meta-policy πmeta, we +use convolutional layers (Fukushima & Miyake, 1982; +Krizhevsky et al., 2017) to extract features from the Karel +states and then process them with GRU layers for predict- +ing program embeddings. To optimize the meta-policy, we +use two popular reinforcement learning algorithms, PPO +(Schulman et al., 2017) and SAC (Haarnoja et al., 2018), +and report their experimental results as HPRL-PPO and +HPRL-SAC, respectively. +More details on hyperparameters, training procedure, and +implementation can be found in Section C. +5.2.3 +Baseline Approaches +To comprehensively understand the efficacy of the proposed +approach, we compare HPRL with the following baselines. +DRL and DRL-abs. Deep RL baselines from (Trivedi et al., +2021). DRL observes a raw state (grids) input from the +Karel environment, while DRL-abs is a recurrent neural net- +work policy that takes abstracted state vectors from the en- +vironment as input. The abstracted state vectors consists of +binary value of the current state (e.g. [frontIsClear() +== True, markerPresent()==False, ...]). +VIPER. A programmatic RL method proposed by Bastani +et al. (2018). It uses a decision tree to imitate the behavior +of a learned DRL policy. +LEAPS. A programmatic RL framework proposed by +Trivedi et al. (2021) that uses Cross-Entropy Method (Ru- +binstein, 1997) to search task-solving program in a learned +continuous program embedding space. +LEAPS-ours. The LEAPS framework trained on the pro- +posed Karel program dataset described in Section 5.2.1. +This is used to compare our proposed program dataset +generation procedure with the generation approach used +in (Trivedi et al., 2021). +More details of these baselines can be found in Section A. +5.3 +Experimental Results +We evaluate the performance in terms of cumulative return +of all methods on the KAREL problem set and the KAREL- +HARD problem set. The experimental results are presented +in Table 1 and Table 2, respectively. The range of the cu- +mulative return is within [0, 1] on tasks without penalty, and +within [−1, 1] on tasks with penalty. Section B describes +the detailed definition of the reward function for each task. + +Hierarchical Programmatic Reinforcement Learning +Table 1. Mean return and standard deviation of all methods across the KAREL problem set, evaluated over three random seeds. HPRL-PPO +outperforms all prior approaches and achieves the maximum score on all tasks. HPRL-SAC completely solves five out of six tasks. +Method +STAIRCLIMBER +FOURCORNER +TOPOFF +MAZE +CLEANHOUSE +HARVESTER +DRL +1.00 ± 0.00 +0.29 ± 0.05 +0.32 ± 0.07 +1.00 ± 0.00 +0.00 ± 0.00 +0.90 ± 0.10 +DRL-abs +0.13 ± 0.29 +0.36 ± 0.44 +0.63 ± 0.23 +1.00 ± 0.00 +0.01 ± 0.02 +0.32 ± 0.18 +VIPER +0.02 ± 0.02 +0.40 ± 0.42 +0.30 ± 0.06 +0.69 ± 0.05 +0.00 ± 0.00 +0.51 ± 0.07 +LEAPS +1.00 ± 0.00 +0.45 ± 0.40 +0.81 ± 0.07 +1.00 ± 0.00 +0.18 ± 0.14 +0.45 ± 0.28 +LEAPS-ours +1.00 ± 0.00 +0.75 ± 0.43 +0.76 ± 0.10 +1.00 ± 0.00 +0.32 ± 0.02 +0.70 ± 0.03 +HPRL-SAC +1.00 ± 0.00 +1.00 ± 0.00 +0.36 ± 0.13 +1.00 ± 0.00 +1.00 ± 0.00 +1.00 ± 0.00 +HPRL-PPO +1.00 ± 0.00 +1.00 ± 0.00 +1.00 ± 0.00 +1.00 ± 0.00 +1.00 ± 0.00 +1.00 ± 0.00 +Table 2. Mean return and standard deviation of all methods across +the KAREL-HARD problem set, evaluated over three random seeds. +HPRL-PPO achieves best performance across all tasks. +Method +DOORKEY +ONESTROKE +SEEDER +SNAKE +LEAPS +0.50 ± 0.00 +0.65 ± 0.24 +0.51 ± 0.06 +0.23 ± 0.10 +LEAPS-ours +0.50 ± 0.00 +0.68 ± 0.11 +0.56 ± 0.00 +0.28 ± 0.08 +HPRL-SAC +0.50 ± 0.00 +0.76 ± 0.08 +0.32 ± 0.10 +0.25 ± 0.06 +HPRL-PPO +0.50 ± 0.00 +0.80 ± 0.03 +0.58 ± 0.10 +0.33 ± 0.12 +The performance of DRL, DRL-abs, VIPER, and LEAPS re- +produced with the implementation provided by Trivedi et al. +(2021). The average cumulative return and standard devia- +tion of LEAPS-ours, HPRL-PPO and HPRL-SAC on each +task are evaluated over three random seeds to ensure sta- +tistical significance. The programs synthesized by LEAPS, +LEAPS-ours, and HPRL are presented in Section F. +Overall Karel Task Performance. The experimental re- +sults on Table 1 show that HPRL-PPO outperforms all other +approaches on all tasks. Furthermore, HPRL-PPO can com- +pletely solve all the tasks in the KAREL problem set. We +also observe that LEAPS-ours outperforms LEAPS on five +out of six tasks in the KAREL problem set, showing that +the proposed program generation process helps improve the +quality of the program embedding space and lead to the +better program search result. +Overall Karel-Hard Task Performance. To further testify +the efficacy of the proposed method, we evaluate LEAPS, +LEAPS-ours, HPRL-PPO, and HPRL-SAC on the KAREL- +HARD problem set. HPRL-PPO outperforms other meth- +ods on ONESTROKE, SEEDER, and SNAKE, while all ap- +proaches perform similarly on DOORKEY. The complexity +of ONESTROKE, SEEDER, and SNAKE makes it difficult for +LEAPS to find satisfactory policies from the program em- +bedding space simply by searching. In contrast, HPRL-PPO +addresses this by composing a series of programs to in- +crease the expressiveness and perplexity of the synthesized +program. We also observe that LEAPS-ours achieve better +performance than LEAPS, further justifying the efficacy of +the proposed program generation procedure. +PPO vs. SAC. HPRL-SAC can still deliver competitive +performance in comparison with HPRL-PPO. However, we +find that HPRL-SAC is more unstable on complex tasks +(e.g. TOPOFF) and tasks with additional constraints (e.g. +SEEDER and SNAKE). On the other hand, HPRL-PPO is +more stable across all tasks and achieve better performance +on both problem sets. Hence, we adopt HPRL-PPO as our +main method in the following experiments. +5.4 +Additional Experiments +We +design +experiments +to +justify +(1) +whether +LEAPS (Trivedi et al., 2021) and our proposed framework +can synthesize out-of-distributional programs, (2) the +necessity of the proposed compression encoder and decoder, +and (3) the effectiveness of learning from dense rewards +made possible by the hierarchical design of our framework. +5.4.1 +Synthesizing Out-of-Distributional Programs +Programs that LEAPS can produce are fundamentally lim- +ited by the distribution of the program dataset since it +searches for programs in the learned embedding space. +More specifically, it is impossible for LEAPS to synthesize +programs that are significantly longer than the programs +provided in the dataset. This section aims to empirically +verify this hypothesis and evaluate the capability of generat- +ing out-of-distributional programs. We create a set of target +programs of lengths 25, 50, 75, and 100, each consisting +of primitive actions (e.g. move, turnRight). Then, we +ask LEAPS and HPRL to fit each target program based on +how well the program produced by the two methods can +reconstruct the behaviors of the target program. The recon- +struction performance is calculated as one minus the nor- +malized Levenshtein Distance between the state sequences +from the execution trace of the target program and from the +execution trace of the synthesized program. +The result is presented in Table 3. HPRL consistently outper- +forms LEAPS with varying target program lengths, and the +gap between the two methods grows more significant when +the target program becomes longer. We also observe that +the reconstruction score of LEAPS drops significantly as + +Hierarchical Programmatic Reinforcement Learning +Table 3. Learning +to +synthesize +out-of-distributional +pro- +grams. HPRL demonstrates superior performance in synthesizing +out-of-distributionally long programs compared to LEAPS. The +gap between the two methods grows more significant when the +length of the target program increases. +Method +Program Reconstruction Performance +Len 25 +Len 50 +Len 75 +Len 100 +LEAPS +0.59 (0.14) +0.31 (0.10) +0.20 (0.05) +0.13 (0.08) +HPRL +0.60 (0.03) +0.34 (0.03) +0.29 (0.03) +0.26 (0.02) +Improvement +1.69% +9.68% +45.0% +100.0% +Table 4. Dimensionality of the Program Embedding Space. +The 64-dimensionaly program embedding space demonstrates the +best task performance with satisfactory reconstruction results. +dim(z) +Reconstruction +Task Performance +Program +Execution +CLEANHOUSE +SEEDER +16 +81.70% +63.21% +0.47 (0.06) +0.21 (0.02) +32 +94.46% +86.00% +0.84 (0.27) +0.35 (0.16) +64 +97.81% +95.58% +1.00 (0.00) +0.58 (0.10) +128 +99.12% +98.76% +1.00 (0.00) +0.57 (0.03) +256 +99.65% +99.11% +1.00 (0.00) +0.54 (0.11) +the length of target programs exceeds 40, which is the maxi- +mum program length of the program datasets. This suggests +that HPRL can synthesize out-of-distributional programs. +Note that the performance of HPRL can be further improved +when setting the horizon of the meta-policy |H| to a larger +number. Yet, for this experiment, we fix it to 5 to better +analyze our method. More details about the implementation +and the evaluation metrics can be found in Section E. +5.4.2 +Dimensionality of Program Embedding Space +Learning a higher-dimensional program embedding space +can lead to better optimization in the program reconstruction +loss (Eq. 5) and the latent behavior reconstruction loss (Eq. +4). Yet, learning a meta-policy in a higher-dimensional +action space can be unstable and inefficient. To investigate +this trade-off and verify our contribution of employing the +compression encoder fω and compression decoder gψ, we +experiment with various dimensions of program embedding +space and report the result in Table 4. +The reconstruction accuracy measures if learned encoders +and decoders can perfectly reconstruct an input program +or its execution trace. The program embedding space with +different dimensionalities are also evaluated in terms of +task performance in CLEANHOUSE and SEEDER since they +are considered more difficult. The result indicates that a 64- +dimensional program embedding space achieves satisfactory +reconstruction accuracy and performs the best on the tasks. +Therefore, we take this dimension as the default setting for +our proposed method. +5.4.3 +Learning from Episodic Reward +We design our framework to synthesize a sequence of pro- +grams, allowing for accurately rewarding correct programs +and penalizing wrong programs (i.e., better credit assign- +ment) with dense rewards. In this section, we design ex- +periments to investigate the effectiveness of this design. To +this end, instead of receiving a reward for executing each +program (i.e., dense) in the environment, we modify CLEAN- +HOUSE and SEEDER so that they only return cumulative +rewards after all |H| programs have been executed (i.e., +episodic). The learning performance is shown in Figure +4, demonstrating that learning from dense rewards yields +better sample efficiency compared to learning from episodic +rewards. This performance gain is made possible by the +hierarchical design of HPRL, which can better deal with +credit assignment. In contrast, LEAPS (Trivedi et al., 2021) +is fundamentally limited to learning from episodic rewards. +Figure 4. Learning from Episodic Reward. We compare learn- +ing from dense and episodic rewards in CLEANHOUSE and +SEEDER. Learning from dense rewards achieves better sample +efficiency in both tasks, which is made possible by the hierarchical +design of our proposed framework. +6 +Conclusion +We propose a hierarchical programmatic reinforcement +learning framework, dubbed HRPL, which re-formulates +solving a reinforcement learning task as synthesizing a task- +solving program that can be executed to interact with the +environment and maximize the return. Specifically, we first +learn a program embedding space that continuously param- +eterizes a diverse set of programs sampled from a program +dataset generated based on our proposed program genera- +tion procedure. Then, we train a meta-policy, whose action +space is the learned program embedding space, to produce +a series of programs (i.e., predict a series of actions) to +yield a composed task-solving program. Experimental re- +sults in the Karel domain on two problem sets demonstrate +that our proposed framework consistently outperforms base- +lines by large margins. 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We further compress the latent program space consists of a fully connected linear neural network +with input dimension of 256 and output dimension of [16, 32, 64, 128]. Please note that VAE with 256 dimension does not +include the fully connected linear neural network. The meta policy neural network consists of the CNN neural nework as +state feature extractor and fully-connected linear layer for the action and value branch. The CNN neural network includes +two convolutional layer. The filter size of the first convolutional layer is 32 with 4 channel. The filter size of the second +convolutional layer is 32 with 2 channel. The output of the state embedding is flatten to a vector of the same size as the +output action vector. +The pseudocode of HPRL is shown at Algorithm 1. +Algorithm 1 Hierarchical Programmatic Reinforcement Learning +Input: Program Dataset Dprogram, VAE Training Epoch Nepoch, Meta-Policy Training Step Tmeta, Program Synthesis +Step |H| +Output: Task Solving Program P +1: Initialize the program encoder qφ and decoder pθ, compression encoder fω and decoder gψ, neural execution policy π. +2: for epoch in range(1, Nepoch) do +3: +for program ρ in Dprogram do +4: +z = fω(qφ(ρ)) +5: +ˆρ = pθ(gψ(z)) +6: +compute Ltotal = LP +θ,φ,ω,φ(ρ) + λLL +π(ρ, π) +7: +fit φ, ω, ψ, θ, π to minimize Ltotal +8: +end for +9: end for +10: Initialize a meta-policy πmeta +11: Load and fix pθ, gψ for Meta-Policy Training +12: for Training Episode in range(1, Tmeta/|H|) do +13: +Receive initial state s1 from the Karel environment +14: +for i in range(1, |H|) do +15: +zi = πmeta(si) +16: +ρi = pθ(gψ(zi)) +17: +Interact with the environment by EXEC(ρi) +18: +Receive [si +1, ..., si +T ] and [ri +1, ..., ri +Ti] +19: +ri+1 = �T i +t=1 ri +t +20: +si+1 = [si +1, si +2, ..., si +Ti][−1] +21: +end for +22: +Calculate Jπmeta based on the collected (si, zi, ri+1, si+1) +23: +fit πmeta to maximize Jπmeta +24: end for +A.2 +DRL +The DRL method implements a deep neural network trained on PPO algorithm for 2M timesteps to learn the policy that +takes the raw states (grids) from the Karel environment as input and predict the next action. The raw state is a binary tensor +representing the state of each grid. + +Hierarchical Programmatic Reinforcement Learning +A.3 +DRL-abs +DRL-abs +is +a +deep +neural +network +utilizing +a +recurrent +policy +and +trained +on +PPO +algorithm +due +to +the +better +performance +compared +with +SAC. +It +is +also +trained +for +2M +timesteps. +The +input +is +the abstract state of the Karel environment instead of Karel raw states (grids). +The abstract states +are represented by [frontIsClear() == True, leftIsClear()==False, rightIsClear()==True, +markerPresent()==False, noMarkersPresent()==True], which is a binary vector that describes the cur- +rent state. +A.4 +VIPER +VIPER is a programmatic RL framework proposed by Bastani et al. (2018) that uses a decision tree to imitate the behavior +of a given neural network teacher policy. Bastani et al. (2018) takes the best DRL policy networks as its teacher policy. +Since VIPER is not capable of synthesizing looping behaviors, it can be used to testify other approaches that employ a +program embedding space to synthesize more complex programs. +A.5 +LEAPS +LEAPS is a programmatic RL framework proposed by Trivedi et al. (2021). The training framework includes two stages. +First, it trains a model with an encoder-decoder architecture to learn a continuous program embedding space. The second +stage utilizes the Cross-Entropy Method (Rubinstein, 1997), searching over the leaned program embedding space to optimize +the program policy for each task. +A.6 +LEAPS-ours +LEAPS-ours uses the same framework as LEAPS but trained on our proposed dataset when learning a program embedding +space. +B +Problem Set Details +B.1 +KAREL Problem Set Details +The KAREL problem set introduced in (Trivedi et al., 2021) consists of six different tasks: STAIRCLIMBER, FOURCORNER, +TOPOFF, MAZE, CLEANHOUSE and HARVESTER. The performance of the policy networks is measured by averaging the +rewards of 10 random environment initial configurations. All experiments are tested on 8 × 8 grid except for CLEANHOUSE. +Figure 5 visualizes the ideal end states and one of their random initial configurations of all tasks. +B.2 +STAIRCLIMBER +In this task, the agent is asked to move along the stair to reach the marked grid. The initial location of the agent and the +marker are randomized near the stair with marker on the higher end. The reward is defined as 1 if the agent reaches the +marked grid, and 0 otherwise. +B.3 +FOURCORNER +The goal of the agent is to place a marker on each corner to earn the reward. Once any marker is placed on the wrong +location, the reward is 0. The reward is the number of corrected placed markers multiplied by 0.25. The initial position of +the agent is at the last row of the environment facing east. +B.4 +TOPOFF +The agent is asked to place markers on marked grids and reach the destination on the rightest grid of the bottom row in +this task. The reward is defined as the consecutive correct states of the last rows until the agent puts a marker on an empty +location or do not place a marker on a marked grid. If the agent ends up on the rightest grid of the last row, a bonus reward +is given. The agent is always initiated on the leftist grid of the bottom row while the locations of markers in the last row are +randomized. + +Hierarchical Programmatic Reinforcement Learning +B.5 +MAZE +In this task, the agent has to navigate itself to reach the marked destination. The locations of markers and the agent as well as +the configuration of the maze are randomized. The reward is 1 if the agent successfully reaches the marked grid or otherwise +0. +B.6 +CLEANHOUSE +There is some garbage (markers) around the apartment so the agent is asked to clean them up. The agent will receive more +rewards for collecting more markers on the grid. A grid is of size 14 × 22 which represents an apartment. The location of +the agent is fixed while the marker locations are randomized. The reward is defined as the number of markers picked to +divide the total number of markers in the initial Karel state. +B.7 +HARVESTER +The goal is to collect more markers on the grid, with markers appearing in all grids in the initial Karel environment. The +reward is defined as the number of collected markers divided by the total markers in the initial state. +B.8 +KAREL-HARD Problem Set Details +Since all the tasks in the original Karel benchmark are well-solved by our method, we proposed a newly designed Karel-Hard +benchmark to further evaluate the capability of HPRL. We define the state transition functions and reward functions for +DOORKEY, ONESTROKE, SEEDER and SNAKE based on Karel states. Each task includes more constraints and more +complex structures, e.g. two-phase structure for DOORKEY, the restriction of no revisiting for ONESTROKE. +The performance of the policy networks is measured by averaging the rewards of 10 random environment initial configura- +tions. The range of cumulative reward in all KAREL-HARD tasks is [0.0, 1.0]. Figure 6 visualize the ideal end states and +one of their random initial configurations of all tasks. +B.9 +DOORKEY +A 8 × 8 grid is split into two areas: a 6 × 3 left chamber and a 6 × 2 right chamber. The two chambers are unconnected +in the beginning. The agent has to pick up the marker in the left chamber to unlock the door, and then get into the right +chamber to place the marker on the top of target(marker). The initial location of the agent, the key(marker) in the left room +and the target(marker) in the right room are randomly initialized. The reward is defined as 0.5 for picking up the key and the +other 0.5 for placing the marker on the marked grid. +B.10 +ONESTROKE +The goal is to make the agent traverse all grids without revisiting. The visited grids will become a wall and the episode will +terminate if the agent hit the wall. The reward is defined as the number of grids visited divide by the total empty grids in +initial Karel environment. The initial location of the agent is randomized. +B.11 +SEEDER +The goal is to put markers on each grid in the Karel environment. The episode will end if makers are repeatedly placed. +The reward is defined as the number of markers placed divide the total empty grids in initial Karel environment. The initial +location of the agent is randomized. +B.12 +SNAKE +In this task, the agent acts like the head of the snake and the goal is to eat(pass through) as much food(markers) as possible +without hitting its body. There is always exactly one marker existing in the environment until 20 markers are eaten. Once +the agent pass the marker, the snake body length will increase 1 and one new marker will appear on the other position of the +environment. The rewards is defined as the number of markers eaten divided by 20. The locations that the markers will +appear are fixed, while the initial agent location is randomized. + +Hierarchical Programmatic Reinforcement Learning +C +Hyperparameters and Settings +C.1 +LEAPS +Following the setting of LEAPS(Trivedi et al., 2021), we experiment with sets of hyperparameters when searching the +program embedding space to optimize the reward for both LEAPS and LEAPS-ours. The settings are described in Table 5 +and Table 6. S, σ, # Elites, Exp Decay and DI represent population size, standard deviation, exponential σ decay and initial +distribution, respectively. +Karel-Hard tasks +Table 5. LEAPS experiment settings on KAREL-HARD tasks. +LEAPS +S +σ +# Elites +Exp Decay +DI +DOORKEY +32 +0.25 +0.1 +False +N(0, 0.1Id) +ONESTROKE +64 +0.5 +0.05 +True +N(1, 0) +SEEDER +32 +0.25 +0.1 +False +N(0, 0.1Id) +SNAKE +32 +0.25 +0.2 +False +N(0, Id) +Reconstruction tasks +Table 6. LEAPS experiment settings on Program Reconstruction tasks. +LEAPS +S +σ +# Elites +Exp Decay +DI +Len 25 +32 +0.5 +0.05 +True +N(0, Id) +Len 50 +32 +0.5 +0.2 +True +N(0, 0.1Id) +Len 75 +64 +0.5 +0.05 +True +N(0, 0.1Id) +Len 100 +64 +0.5 +0.1 +True +N(0, 0.1Id) +C.2 +LEAPS-ours +Karel tasks +Table 7. LEAPS-ours experiment settings on KAREL tasks. +LEAPS-ours +S +σ +# Elites +Exp Decay +DI +STAIRCLIMBER +32 +0.5 +0.05 +True +N(0, 0.1Id) +FOURCORNERS +32 +0.5 +0.1 +True +N(1, 0) +TOPOFF +64 +0.25 +0.05 +Trie +N(0, 0.1Id) +MAZE +64 +0.1 +0.2 +False +N(1, 0) +CLEANHOUSE +64 +0.1 +0.05 +False +N(1, 0) +HARVESTER +64 +0.5 +0.05 +True +N(1, 0) +Karel-Hard tasks +Table 8. LEAPS-ours experiment settings on KAREL-HARD tasks. +LEAPS-ours +S +σ +# Elites +Exp Decay +DI +DOORKEY +64 +0.5 +0.2 +True +N(1, 0) +ONESTROKE +64 +0.5 +0.05 +False +N(0, Id) +SEEDER +32 +0.25 +0.1 +False +N(1, 0) +SNAKE +32 +0.25 +0.05 +False +N(0, 0.1, Id) + +Hierarchical Programmatic Reinforcement Learning +Table 9. Hyperparameters of HPRL-PPO and HPRL-SAC Training +Training +Settngs +SAC +PPO +Max # Subprogram +5 +5 +Max Subprogram Length +40 +40 +Batch Size +1024 +256 +Specific Parameters +Init. Temperatur: 0.0002 +Actor Update Frequency: 200 +Critic Target Update Frequency: 200 +Num Seed Steps: 20000 +Reply Buffer Size: 5M +Training Steps: 25M +Alpha Learning Rate: 0.0001 +Actor Learning Rate: 0.0001 +Critic Learning Rate: 0.00001 +β: [0.9, 0.999] +Critic τ: 0.005 +Number of parallel actors: 16 +Discount factor: 0.99 +Q-critic Hidden Dimension: 16 +Learning Rate: 0.00005 +Entropy Coefficient: 0.1 +Rollout Size: 12800 +Eps: 0.00001 +α: 0.99 +γ: 0.99 +Use GAE: True +GAE lambda: 0.95 +Value Loss Coefficient: 0.5 +Clip Param: 0.2 +max grad. norm.: 0.5 +Update Epoch: 3 +clip param.: 0.2 +Training Steps: 25M +Table 10. The statistical distribution of programs containing each token in our generated dataset. +IFELSE +IF +WHILE +REPEAR +Our Dataset +41% +47% +54% +22% +C.3 +HPRL +Pretraining VAE +• Latent embedding size: 64 +• GRU Hidden Layer Size: 256 +• # GRU layer for encoder/decoder: 1 +• Batch Size: 256 +• Nonlinearity: Tanh() +• Optimizer: Adam +• Learning Rate: 0.001 +• Latent Loss Coefficient (β): 0.1 +RL training on Meta Policies +The Hyperparameters for HPRL-PPO and HPRL-SAC training are reported in Table 9. For each task, we test on 3 different +random seeds and take the average to measure the performance. +D +The KAREL Program Datasets Generation +The Karel program dataset used in this work include 1 million program sequence, with 85% as training dataset and 15% +as evaluation dataset. In addition to sequences of program tokens, the KAREL program dataset also include execution +demonstrations (e.g., state transition and action sequence) of each program in the dataset, which can be used for the latent +behavior reconstruction loss described in Section 4.1. +To further improve the data quality, we added some heuristic rules while selecting data to filter out the programs with +repetitive or offsetting behavior. The unwanted programs that we drop while collecting data are mainly determined by the +following rules: + +Hierarchical Programmatic Reinforcement Learning +• Contradictory Primitive Actions: turnLeft followed by turnRight, pickMarker followed by putMarker, or +vice versa. +• Meaningless Programs: end_state == start_state after program execution +• Repetitive behaviors: a program which has the longest common subsequence of tokens longer than 9 +We further analyze the distribution of the generated program sequences based on the control flow (e.g., IF, IFELSE) +and loop command (e.g., WHILE, REPEAT). The statistical probabilities of programs containing control flow or loop +command are listed in Table 10. Results show that more than 40% of the programs in the collected program sequences +contain at least one of the control of loop command, ensuring the diversity of the generated programs. +E +More on Learning to Synthesize Out-of-distributional Programs +Measure the Performance. To measure the performance of programs synthesized by different methods, we first collect +execute each target program, yielding a target state sequence τtarget = [s1, s2, . . . , sTtarget]. Then, we reset the Karel +environment to the initial state s1. For our proposed framework, we synthesize a sequence of programs with the following +procedure and optimize a program reconstruction reward to match the target program. As described in Section 4.3, at +each macro training time step n, 1 ≤ n ≤ |H|, we collect the state sequence τP = [sP +1 , sP +2 , . . . , sP +TP] from the executing +of task-solving program P = ⟨ρi|i = 1, .., n⟩, and calculate the program reconstruction reward rn = 1 − D(τtarget, τP) +where D is the normalized Levenshtein distance. For executing the programs synthesized by LEAPS and LEAPS-ours, +we simply start executing programs after resetting the Karel environment to the initial state s1 and calculate the return. A +Python implementation that calculates the program reconstruction performance is as follows. +1 +import numpy as np +2 +3 +def _compare_demos(demo1, demo1_len, demo2, demo2_len): +4 +if demo2_len == 0: return 0 +5 +if demo1_len == 1 and demo2_len == 1: return 1 +6 +distances = np.zeros((demo1_len + 1, demo2_len + 1)) +7 +for t1 in range(demo1_len + 1): +8 +distances[t1][0] = t1 +9 +for t2 in range(demo2_len + 1): +10 +distances[0][t2] = t2 +11 +a = 0 +12 +b = 0 +13 +c = 0 +14 +for t1 in range(1, demo1_len + 1): +15 +for t2 in range(1, demo2_len + 1): +16 +if np.array_equal(demo1[t1-1], demo2[t2-1]): +17 +distances[t1][t2] = distances[t1 - 1][t2 - 1] +18 +else: +19 +a = distances[t1][t2 - 1] +20 +b = distances[t1 - 1][t2] +21 +c = distances[t1 - 1][t2 - 1] +22 +if (a <= b and a <= c): +23 +distances[t1][t2] = a + 1 +24 +elif (b <= a and b <= c): +25 +distances[t1][t2] = b + 1 +26 +else: +27 +distances[t1][t2] = c + 1 +28 +return 1.0 - ((distances[demo1_len][demo2_len]) / (max(demo1_len, demo2_len)-1)) +F +Synthesized Programs +In this section, we provide qualitative results (i.e., synthesized programs) of our proposed framework (HPRL-PPO), LEAPS, +and LEAPS-ours. The programs synthesized for the tasks in the KAREL problem set are shown in Figure 7 (STAIRCLIMBER, +TOPOFF, and CLEANHOUSE), Figure 8 (FOURCORNER, MAZE), and Figure 9 (HARVESTER). The programs synthesized +for the tasks in the KAREL-HARD problem set are shown in Figure 10 (DOORKEY), Figure 11(ONESTROKE), and Figure +12 (SEEDER and SNAKE). + +Hierarchical Programmatic Reinforcement Learning +stairClimber +(a) STAIRCLIMBER +fourCorners +(b) FOURCORNER +topOff +(c) TOPOFF +maze +(d) MAZE +harvester +(e) HARVESTER +cleanHouse +(f) CLEANHOUSE +Figure 5. Illustrations of the initial and desired final state of each task in the KAREL Problem set introduced in by Trivedi et al. (2021). +Note that these illustrations are from (Trivedi et al., 2021). The position of markers, walls, and agent’s position are randomly set according +to the configurations of each tasks. More details are provided in Section B.1. + +. +. +. +. +. +. +. +. +. +. +. +. +.. +. +. +. +. +. +. +一 +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +.Hierarchical Programmatic Reinforcement Learning +(a) DOORKEY +(b) ONESTROKE +(c) SEEDER +(d) SNAKE +Figure 6. Illustrations of the initial and final state of each task in the proposed KAREL-HARD Problem Set. The position of markers, walls, +and agent’s position are randomly set according to the configurations of each tasks. More details are provided in Section B.8. + +CHierarchical Programmatic Reinforcement Learning +Karel Programs +STAIRCLIMBER +LEAPS +DEF run m( +WHILE c( noMarkersPresent c) w( +turnRight +move +w) +WHILE c( rightIsClear c) w( +turnLeft +w) +m) +LEAPS-ours +DEF run m( +turnRight +turnRight +WHILE c( noMarkersPresent c) w( +turnRight +move +w) +m) +HPRL-PPO +DEF run m( +WHILE c( noMarkersPresent c) w( +turnRight +move +turnRight +move +w) +m) +TOPOFF +LEAPS +DEF run m( +WHILE c( noMarkersPresent c) w( +move +w) +putMarker +move +WHILE c( not c( markersPresent c) +c) w( +move +w) +putMarker +move +WHILE c( not c( markersPresent c) +c) w( +move +w) +putMarker +move +turnRight +turnRight +turnRight +turnRight +turnRight +turnRight +turnRight +turnRight +m) +LEAPS-ours +DEF run m( +WHILE c( not c( rightIsClear c) c +) w( +WHILE c( not c( +markersPresent c) c) w( +move +w) +putMarker +move +w) +WHILE c( not c( rightIsClear c) c +) w( +pickMarker +w) +m) +HPRL-PPO +DEF run m( +move +move +REPEAT R=5 r( +move +WHILE c( noMarkersPresent c) +w( +move +w) +putMarker +r) +m) +CLEANHOUSE +LEAPS +DEF run m( +WHILE c( noMarkersPresent c) w( +turnRight +move +move +turnLeft +turnRight +pickMarker +w) +turnLeft +turnRight +m) +LEAPS-ours +DEF run m( +move +WHILE c( noMarkersPresent c) w( +turnRight +move +WHILE c( frontIsClear c) w( +move +pickMarker +w) +w) +m) +HPRL-PPO +DEF run m( +REPEAT R=4 r( +REPEAT R=4 r( +REPEAT R=4 r( +turnRight +move +pickMarker +move +r) +r) +r) +m) +DEF run m( +REPEAT R=4 r( +REPEAT R=4 r( +REPEAT R=4 r( +REPEAT R=4 r( +turnRight +move +pickMarker +move +r) +r) +r) +r) +m) +DEF run m( +REPEAT R=4 r( +REPEAT R=4 r( +REPEAT R=4 r( +pickMarker +move +turnRight +move +r) +r) +r) +m) +Figure 7. Example programs on Karel tasks: STAIRCLIMBER, TOPOFF and CLEANHOUSE. The programs with best rewards out +of all random seeds are shown. + +Hierarchical Programmatic Reinforcement Learning +FOURCORNER +LEAPS +DEF run m( +turnRight +move +turnRight +turnRight +turnRight +WHILE c( frontIsClear c) w( +move +w) +turnRight +putMarker +WHILE c( frontIsClear c) w( +move +w) +turnRight +putMarker +WHILE c( frontIsClear c) w( +move +w) +turnRight +putMarker +WHILE c( frontIsClear c) w( +move +w) +turnRight +putMarker +m) +LEAPS-ours +DEF run m( +REPEAT R=5 r( +WHILE c( frontIsClear c) w( +move +w) +IFELSE c( not c( rightIsClear +c) c) i( +turnLeft +putMarker +i) +ELSE e( +putMarker +e) +r) +m) +HPRL-PPO +DEF run m( +move +move +WHILE c( frontIsClear c) w( +move +w) +turnLeft +putMarker +m) +DEF run m( +move +move +WHILE c( frontIsClear c) w( +move +w) +putMarker +WHILE c( frontIsClear c) w( +move +w) +turnLeft +m) +DEF run m( +move +move +WHILE c( frontIsClear c) w( +move +w) +putMarker +turnLeft +putMarker +WHILE c( frontIsClear c) w( +move +w) +putMarker +putMarker +m) +MAZE +LEAPS +DEF run m( +IF c( frontIsClear c) i( +turnLeft +i) +WHILE c( noMarkersPresent c) w( +turnRight +move +w) +m) +LEAPS-ours +DEF run m( +turnRight +turnRight +WHILE c( noMarkersPresent c) w( +turnRight +move +w) +m) +HPRL-PPO +DEF run m( +WHILE c( noMarkersPresent c) w( +move +turnRight +w) +move +m) +Figure 8. Example programs on Karel tasks: FOURCORNER and MAZE. The programs with best rewards out of all random seeds +are shown. + +Hierarchical Programmatic Reinforcement Learning +HARVESTER +LEAPS +DEF run m( +turnLeft +turnLeft +pickMarker +move +pickMarker +pickMarker +move +pickMarker +move +pickMarker +move +pickMarker +move +turnLeft +pickMarker +move +pickMarker +move +pickMarker +move +pickMarker +move +turnLeft +pickMarker +move +pickMarker +move +pickMarker +move +pickMarker +move +turnLeft +pickMarker +move +pickMarker +move +pickMarker +move +m) +LEAPS-ours +DEF run m( +WHILE c( leftIsClear c) w( +REPEAT R=4 r( +pickMarker +move +r) +turnLeft +pickMarker +move +turnLeft +pickMarker +move +w) +turnLeft +pickMarker +turnLeft +m) +HPRL-PPO +DEF run m( +REPEAT R=4 r( +REPEAT R=4 r( +pickMarker +turnRight +move +pickMarker +turnRight +move +pickMarker +move +pickMarker +move +r) +turnRight +pickMarker +move +pickMarker +move +pickMarker +move +r) +m) +Figure 9. Example programs on Karel tasks: HARVESTER. The programs with best rewards out of all random seeds are shown. + +Hierarchical Programmatic Reinforcement Learning +Karel-Hard Programs +DOORKEY +LEAPS +DEF run m( +move +turnRight +putMarker +pickMarker +move +WHILE c( leftIsClear c) w( +pickMarker +move +w) +m) +LEAPS-ours +DEF run m( +WHILE c( rightIsClear c) w( +turnRight +pickMarker +turnLeft +pickMarker +pickMarker +pickMarker +pickMarker +move +turnLeft +move +w) +m) +HPRL-PPO +DEF run m( +REPEAT R=4 r( +REPEAT R=4 r( turnRight move +pickMarker move +pickMarker move r) +pickMarker move r) +m) +DEF run m( +REPEAT R=5 r( +turnRight move +REPEAT R=5 r( move r) +move pickMarker move +r) +m) +DEF run m( +REPEAT R=4 r( +REPEAT R=4 r( +turnRight move +REPEAT R=3 r( +move pickMarker +move pickMarker r) +r) +r) +m) +DEF run m( +REPEAT R=4 r( +REPEAT R=4 r( +turnRight move +pickMarker move +pickMarker +REPEAT R=2 r( +pickMarker move +pickMarker pickMarker +r) +r) +r) +m) +DEF run m( +REPEAT R=4 r( +turnRight +REPEAT R=4 r( +turnRight move move +pickMarker move +r) +move pickMarker move +r) +move pickMarker +m) +Figure 10. Example programs on Karel-Hard tasks: DOORKEY. The programs with best rewards out of all random seeds are shown. + +Hierarchical Programmatic Reinforcement Learning +ONESTROKE +LEAPS +DEF run m( +REPEAT R=9 r( +turnRight +turnRight +WHILE c( frontIsClear c) w( +move +w) +turnRight +WHILE c( frontIsClear c) w( +move +w) +r) +turnRight +m) +LEAPS-ours +DEF run m( +turnRight +WHILE c( frontIsClear c) w( +WHILE c( frontIsClear c) w( +WHILE c( frontIsClear c) +w( +WHILE c( frontIsClear +c) w( +move +w) +turnRight +w) +turnRight +w) +turnRight +w) +turnRight +m) +HPRL-PPO +DEF run m( +WHILE c( frontIsClear c) w( move +w) turnRight +WHILE c( frontIsClear c) w( move +w) turnRight +WHILE c( frontIsClear c) w( move +w) turnRight +m) +DEF run m( +WHILE c( frontIsClear c) w( move +w) turnRight +WHILE c( frontIsClear c) w( move +w) turnRight +WHILE c( frontIsClear c) w( move +w) turnRight +m) +DEF run m( +WHILE c( frontIsClear c) w( move +w) turnRight +WHILE c( frontIsClear c) w( move +w) turnRight +WHILE c( frontIsClear c) w( move +w) turnRight +WHILE c( frontIsClear c) w( move +w) turnRight +m) +DEF run m( +WHILE c( frontIsClear c) w( move +w) turnRight +WHILE c( frontIsClear c) w( move +w) turnRight +WHILE c( frontIsClear c) w( move +w) turnRight +m) +Figure 11. Example programs on Karel-Hard tasks: ONESTROKE. The programs with best rewards out of all random seeds are +shown. + +Hierarchical Programmatic Reinforcement Learning +SEEDER +LEAPS +DEF run m( +WHILE c( noMarkersPresent c) w( +turnRight +putMarker +move +move +w) +turnRight +turnRight +turnRight +turnRight +turnRight +turnRight +turnRight +turnRight +m) +LEAPS-ours +DEF run m( +WHILE c( noMarkersPresent c) w( +putMarker +move +turnRight +move +w) +turnRight +turnRight +turnRight +turnRight +turnRight +turnRight +turnRight +turnRight +m) +HPRL-PPO +DEF run m( +putMarker move +putMarker move +putMarker move +putMarker move +putMarker move +turnRight move +m) +DEF run m( +putMarker move +putMarker move +putMarker move +putMarker move +putMarker move +turnRight move +putMarker move +m) +DEF run m( +putMarker move +putMarker move +putMarker move +putMarker move +turnRight move +putMarker move +turnRight move +m) +DEF run m( +putMarker move +putMarker move +putMarker move +putMarker move +turnRight move +putMarker move +turnRight move +DEF run m( +putMarker move +putMarker move +putMarker move +putMarker move +turnRight move +putMarker move +turnRight move +m) +SNAKE +LEAPS +DEF run m( +turnRight +turnLeft +pickMarker +move +move +move +WHILE c( rightIsClear c) w( +turnLeft +move +move +w) +turnLeft +turnLeft +turnLeft +turnLeft +m) +LEAPS-ours +DEF run m( +move +turnRight +pickMarker +pickMarker +WHILE c( rightIsClear c) w( +turnLeft +move +move +w) +turnRight +move +move +move +m) +HPRL-PPO +DEF run m( +move +WHILE c( noMarkersPresent c) w( +move +move +turnLeft +w) +move +turnLeft +m) +DEF run m( +move +WHILE c( noMarkersPresent c) w( +move +move +turnLeft +w) +m) +DEF run m( +move +WHILE c( noMarkersPresent c) w( +move +move +turnLeft +w) +move +turnLeft +m) +Figure 12. Example programs on Karel-Hard tasks: SEEDER and SNAKE. The programs with best rewards out of all random seeds +are shown. + diff --git a/PdFOT4oBgHgl3EQf4jQq/content/tmp_files/load_file.txt b/PdFOT4oBgHgl3EQf4jQq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..131c0ac92a03b86f69d2bb7d3c1075a4e19b5b0e --- /dev/null +++ b/PdFOT4oBgHgl3EQf4jQq/content/tmp_files/load_file.txt @@ -0,0 +1,2152 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFOT4oBgHgl3EQf4jQq/content/2301.12950v1.pdf,len=2151 +page_content='Hierarchical Programmatic Reinforcement Learning via Learning to Compose Programs Guan-Ting Liu * 1 En-Pei Hu * 1 Pu-Jen Cheng 1 Hung-Yi Lee 1 Shao-Hua Sun 1 Abstract Aiming to produce reinforcement learning (RL) policies that are human-interpretable and can gen- eralize better to novel scenarios, Trivedi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFOT4oBgHgl3EQf4jQq/content/2301.12950v1.pdf'} +page_content=' (2021) present a method (LEAPS) that first learns a program embedding space to continuously pa- rameterize diverse programs from a pre-generated program dataset, and then searches for a task- solving program in the learned program embed- ding space when given a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFOT4oBgHgl3EQf4jQq/content/2301.12950v1.pdf'} +page_content=' Despite encour- aging results, the program policies that LEAPS can produce are limited by the distribution of the program dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFOT4oBgHgl3EQf4jQq/content/2301.12950v1.pdf'} +page_content=' Furthermore, during searching, LEAPS evaluates each candidate program solely based on its return, failing to precisely reward correct parts of programs and penalize incorrect parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFOT4oBgHgl3EQf4jQq/content/2301.12950v1.pdf'} +page_content=' To address these issues, we propose to learn a meta-policy that composes a series of pro- grams sampled from the learned program embed- ding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFOT4oBgHgl3EQf4jQq/content/2301.12950v1.pdf'} +page_content=' By composing programs, our pro- posed method can produce program policies that describe out-of-distributionally complex behav- iors and directly assign credits to programs that induce desired behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFOT4oBgHgl3EQf4jQq/content/2301.12950v1.pdf'} +page_content=' We design and con- duct extensive experiments in the Karel domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFOT4oBgHgl3EQf4jQq/content/2301.12950v1.pdf'} +page_content=' The experimental results show that our proposed framework outperforms baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFOT4oBgHgl3EQf4jQq/content/2301.12950v1.pdf'} +page_content=' The ablation studies confirm the limitations of LEAPS and jus- tify our design choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFOT4oBgHgl3EQf4jQq/content/2301.12950v1.pdf'} +page_content=' 1 Introduction Deep reinforcement learning (DRL) leverages the recent advancement in deep learning by reformulating the rein- forcement learning problem as learning policies or value functions parameterized by deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFOT4oBgHgl3EQf4jQq/content/2301.12950v1.pdf'} +page_content=' DRL has achieved tremendous success in various domains, in- cluding controlling robots (Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFOT4oBgHgl3EQf4jQq/content/2301.12950v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFOT4oBgHgl3EQf4jQq/content/2301.12950v1.pdf'} +page_content=' Ibarz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFOT4oBgHgl3EQf4jQq/content/2301.12950v1.pdf'} +page_content=', Equal contribution 1National Taiwan University, Taipei, Tai- wan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFOT4oBgHgl3EQf4jQq/content/2301.12950v1.pdf'} +page_content=' Correspondence to: Shao-Hua Sun 휏2 > 휏3 are principal components of the +deviatoric stress. +3.1.4. Evaluation of horizontal tectonic stress and +dilatancy cofficient +We evaluate the dilatancy coefficient Λ and horizontal +stress leading to formation of a fault inclined by angle 휓 +under certain parameters of rock strength, namely, cohesion +푐 and internal friction coefficient 훼. +Mohr-Coulomb limiting condition is formulated for +fluid-saturated porous medium as follows: +푇 + 훼휎 = 푐 − 훼푝푓, +(17) +where 푝푓 is the pore pressure. +We consider the elastic rock layer located at the depth ℎ +under vertical stress 휎33 and horizontal stresses +휎11 = 휎푒 +11 + 휎푡 +11, +휎22 = 휎푒 +22 + 휎푡 +22, +(18) +due to lateral rock repulsion 휎푒 +11 and 휎푒 +22 as well as tectonic +stresses 휎푡 +11, 휎푡 +22. +Total horizontal stresses due to lateral repulsion in elastic +fluid-saturated layer are expressed as follows (Eaton (1969)): +휎푒 +11 = 휎푒 +22 = +휈 +1 − 휈 휎33 − 푝푓 +1 − 2휈 +1 − 휈 +(19) +Expression (19) was obtained in the absence of thermal- +induced stresses in the rock formation. Temperature of the +rock increases with an increase in the depth according to +geothermal gradient, which depends on rock composition, +thermal conductivity and density of heat flux. Usually +geothermal gradient takes values in the range between 0.5 +◦C up to 20 ◦C with the average value of 3 ◦C for 100m +depth. +Constitutive relation for a heated elastic layer has the +following form (Timoshenko and Goodier, 1970) +휎푖푗 = 2퐺휀푖푗 + 2퐺휈 +1 − 2휈 휀훿푖푗 − 2휅퐺(1 + 휈) +1 − 2휈 +푇푓훿푖푗 +(20) +where 푇푓 is the temperature and 휅 is the thermal expansion +coefficient. +Following the derivation of expressions (19) we consider +elastic half-space with temperature distribution along the +depth 푇푓 += 푇푓(푥3). In this case, the deformations are +expressed as follows +휀11 = 휀22 = 0, +휀33 = 휅푇푓 +1 + 휈 +1 − 휈 +(21) +so that equilibrium equations 휎푖푗,푗 = 0 are satisfied. +Now stress components according to (20) are expressed +as follows: +휎11 = 휎22 = −2휅푇푓퐺1 + 휈 +1 − 휈 , +휎33 = 0. +(22) +Expressions (22) allow to generalize Eaton expressions +(19) for horizontal stresses in a heated fluid-saturated half- +space as follows: +휎푒 +11 = 휎푒 +22 = +휈 +1 − 휈 휎33 − (푝푓 + 푝푡)1 − 2휈 +1 − 휈 , +(23) +Preprint submitted to Journal of Natural Gas Science and Engineering +Page 7 of 26 + +Distributed +Caombined +Conduit +low +ECOTC +high +Conduif-Barrier +higl +Acerctionary +high +Dixic Valley +Prisins +Fault +Damage +Permeability Structures +Lunage +In Fault Zones +2u07 +1 +Shawuogunk +San Gabnel +low +Mountains +Cataclasites +low +Localized +Locatized +Conduit +low ++ Core +high +Barrier300 +Quartzite +250 +K +Axial stress (MPa) +200 +150 +100 +Axial strain + - Radial strain +50 +Volumetric strain +0 +-0.010 +-0.005 +0.000 +0.005 +0.010 +StrainGeomechanical risks of CO2 storage +푝푡 = 2휅푇푓퐺 1 + 휈 +1 − 2휈 +The obtained stress components (23) allow to calculate +the stress intensity 푇 and mean stress 휎 under the applied +tectonic stress 휎푡 +11 and 휎푡 +22 (see expressions in Appendix +A below Eq. (47) and Eq. (18)). Parameters 푇 and 휎 are +substituted into limiting condition (17) and assuming that +휎푡 +22 = 푚휎푡 +11 we find the critical horizontal stress 휎푡 +11(푐푟), at +which the horizontal layer turns into inelastic state. Next, +assuming that the localization of deformations is formed at +the stress 휎푡 +11(푐푟), consider expression (16). Substituting all +the known parameters and the angle of fault inclination 휓, +we find the corresponding dilatancy coefficient Λ. +3.1.5. Evaluation of rock permeability in the damage +zone +The calculated dilatancy coefficient allows one to deter- +mine the increase in permeability of rock damage zone in +the vicinity of the tectonic fault due to plastic deformations +along its plane. Consider the expression for permeability of +the fracture network (Basniev et al., 1993): +푘푓 = +푚푓훿2 +12 , +(24) +where 훿 is the mean fracture aperture, 푚푓 is the fracture +porosity, which is the ratio of the volume of fractures to the +total rock volume. These parameters are related with each +other by the following expression: +푚푓 = 퐷훿, +(25) +where 퐷 is the fracture intensity determined experimentally +as the ratio of total fracture length to the rock cross-section +area. +According to the definition of dilatancy coefficient Λ the +following expression holds +휀푝 = ΛΓ푝, +(26) +where Γ푝 is the intensity of plastic shear deformations (the +second invariant of the shear plastic deformation). +We assume that the ratio of volume of fractures opened +due to rock dilatancy to the geometric volume of the rock +is equal to an increase in the rock volume due to inelastic +deformations 휀푝. Therefore, according to Eq. (26), we can +formulate the following expression for 푚푓: +푚푓 = ΛΓ푝. +(27) +Substituting (27) into (24) and using Eq. (25) we find +푘푓 = (휀푝)3 +12퐷2 = (ΛΓ푝)3 +12퐷2 . +(28) +Expression (28) allows one to determine an increase in +the permeability of near fault damage zone in the presence +of plastic deformations. Note that all the parameters required +to use the obtained expression are determined via standard +laboratory measurements of rock mechanical properties. +The permeability alteration model described above is +embedded into the framework of coupled hydro-geomechanical +model (Section 2). Plastic deformations due to change in +stress state of the rock during CO2 injection are calculated +directly using mechanical simulator FLAC3D with the help +of the relation: +휀푝 = 휀푣 − 휀푒, +휀푒 = Δ휎′ +퐾 , +where 휀푒 are elastic deformations, 휀푣 is the total volumetric +deformation calculated using FLAC3D, Δ휎′ is the change +in mean effective stress between the current and initial rock +state, and 퐾 is the bulk modulus. The fracture intensity 퐷 +varies between 10 and 50 m−1 (Golf-Rakht, 1986), and in +the simulations shown below we assume 퐷 = 30 m−1, which +corresponds to sandstone. +3.1.6. Effect of shear slip on permeability of fractures +closed on natural asperities +Modeling of a rock as elastoplastic medium with internal +friction and dilatancy allows determining deconsolidation +of the fault zone due to shear deformations as described +above (see Eq. (28)). This approach can be used to describe +the alteration of the fault dynamic influence zone (damage +zone) containing microfractures, while it does not allow +calculating the aperture of the main crack in the fault core. +Major fracture opening due to shear deformations is a +result of applied shear stress being larger as compared to +the friction force and resistant force of interaction of natural +asperities as shown in Fig. 5. +Figure 5: Shear slip along the fracture plane, after Barton and +Choubey (1977). +In the framework of Barton-Bandis model (Barton and +Choubey, 1977), fracture aperture due to slip 퐸푑 is expressed +as follows: +퐸푑 = 푈푠 ⋅ tg(푑푚). +(29) +where 푈푠 is the slip displacement, and 푑푚 is the dynamic +angle of dilatancy. Instead of the tangent of 푑푚, we utilize +the dilatancy coefficient Λ = tg(푑푚). +Laboratory experiments on rock samples show that the +dilatancy angle depends on the normal stress, limiting shear +stress, friction coefficient at the walls and residual friction +angle. Zhigulskii and Tikhotskii (2020) show that 3D ge- +omechanical modeling allows evaluating mechanical and +Preprint submitted to Journal of Natural Gas Science and Engineering +Page 8 of 26 + +a) +T=0 +b) +T≠0Geomechanical risks of CO2 storage +hydraulic fracture aperture based on the rock stress state +parameters and shear deformations. +Consider the plane problem of linear fracture in an +elastic orthotropic medium. The fracture of the length 2푎 is +oriented along the principal axes of anisotropy and is loaded +at infinity with horizontal 푝1, vertical 푝3 and tangential 휏 +stresses (see Fig. 6a). Fracture walls are confined with the +stress 푝3 and interact according to dry friction law. Friction +coefficient in static state 훼푢푝 is maximum. If the shear stress +increases up to its critical value 휏푢푝 = 훼푢푝푝3 + 푐, where 푐 +is the cohesion, then the fracture walls start to move and +friction coefficient drops to the value 훼푑푤. As a result, the +shear stress decreases to the value 휏푑푤 = 훼푑푤푝3 and under +the applied stress Δ휏 (see Fig. 6b) +Δ휏 = 휏푢푝 − 휏푑푤 +(30) +the fracture walls are displaced by 푈푠 = 푢1(푥1, 0). +Figure 6: Plane rock domain containing fracture (a) with the +walls interacting according to dry friction law (b). +We define the shear deformation following the study +Garagash and Osiptsov (2021). Equilibrium conditions are +formulated as follows +휎11,1 + 휎13,3 = 0, +휎13,1 + 휎33,3 = 0 +(31) +Consider the function of stresses 퐹 satisfying the follow- +ing conditions +휎11 = 퐹,33, +휎33 = 퐹,11, +휎13 = −퐹,13 +(32) +Under the conditions (32), equilibrium conditions (31) +are satisfied identically. +Strain compatibility condition is formulated as follows +휀11,33 + 휀33,11 = 2휀13,13 +(33) +The condition (33) can be satisfied by using the consti- +tutive relations for transversal isotropic body at plane strain +state 휀22 = 0: +휎11 = 퐶11휀11 + 퐶13휀33, 휎33 = 퐶13휀11 + 퐶33휀33, (34) +휎13 = 퐶44휀13 +The stiffness tensor components for isotropic medium +have the following form: +퐶11 = 퐶33 = 4퐺 1 − 휈 +1 − 2휈 , 퐶12 = 퐶13 = +2퐺 +1 − 2휈 , 퐶44 = 퐺. +Solving (34) with respect to deformations we obtain the +following expressions +휀11 = 푆11휎11 + 푆13휎33, 휀13 = 푆44휎13, +(35) +휀33 = 푆33휎33 + 푆13휎11, +where +푆11 = 퐶33Δ−1 +휀 , 푆33 = 퐶11Δ−1 +휀 , 푆13 = −퐶13Δ−1 +휀 , +푆44 = 퐶−1 +44 , Δ휀 = 퐶11퐶33 − 퐶2 +13. +(36) +Substituting Eqs. (35) into conditions (33) we obtain +푆11퐹,3333 + (2푆13 + 푆44)퐹,1133 + 푆33퐹,1111 = 0. (37) +Eq. (37) can be solved using Fourier transformation +(Nowacki, 1975), and the following expression for displace- +ment along the fracture axis at |푥1| ≤ 푎 is obtained (Gara- +gash and Osiptsov (2021)): +푢1(푥1, 0) = 0.5푎Δ휏(푘1 + 푘2)푆11 +√ +1 − (푥1∕푎)2, (38) +푢3(푥1, 0) = −Δ휏 +(√ +푆11푆33 + 푆13 +) +푥1, +where +푘2 +1,2 = 2푆13+푆44 ± +√ +(2푆13+푆44)2 − 4푆33푆11 +2푆11 +. +Finally substituting displacement 푢1(푥1, 0) into expres- +sion (29) we find the fracture aperture 퐸푑: +퐸푑(푥1) = 푢1(푥1, 0) ⋅ Λ. +(39) +In the mechanical model, we utilize 퐸∗ +푑 value corre- +sponding to the averaged displacement profile 푢1(푥1, 0) +along the fracture 푥1 ∈ [−푎, 푎]: +퐸∗ +푑 = 휋 +4 푎2Δ휏(푘1 + 푘2)푆11Λ. +(40) +By using the fracture aperture determined by Eq. (40) we +calculate the permeability of the main fracture in the fault +core closed on natural asperities as follows: +푘푐 = +푤2 +푐 +12 , +(41) +where 푤푐 is the hydraulic aperture of the main fracture +determined according to Barton and Choubey (1977) as +follows: +푤푐 = +퐸2 +푑 +JRC2.5 , +(42) +where 퐸푑 is in mm. In Eq. (42), JRC is the joint roughness +coefficient determined in the laboratory experiments. We +take JRC = 4 in our study. As it is reported in (Barton and +Choubey, 1977), JRC varies in the range between 1 and 20 +The model of fracture permeability closed on natural as- +perities is embedded into the coupling procedure described +in Section 2 for the description of the activation of the +Preprint submitted to Journal of Natural Gas Science and Engineering +Page 9 of 26 + +a) +b) +木T +2 +up +X3 +△T +X1 +2a +Ui +P1Geomechanical risks of CO2 storage +fault core as follows. If the deformations in the mesh cells +containing fracture core are pure elastic, and the shear stress +휏푛 on the fault plane reaches the critical value: +휏푛 ≥ 훼푢푝|휎′ +푛| + 푐, +(43) +where the shear stress 휏푛 and normal effective stress 휎′ +푛 on +the fault plane with inclination angle 휓 provided plane strain +conditions (see Fig. 7) are given by +휏푛 = (휎푧푧 − 휎푥푥) sin 휓 cos 휓 + 휎푥푧 cos 2휓, +휎′ +푛 = 휎′ +푥푥 cos2 휓 + 휎′ +푧푧 sin2 휓 + 휎푥푧 sin 2휓, +휎′ +푥푥 = 휎푥푥 + 푝, 휎′ +푧푧 = 휎푧푧 + 푝, +then the fracture aperture 퐸∗ +푑 is calculated using Eq. (40). +Stress difference Δ휏 is determined based on the stress state +evaluated by FLAC3D: +Δ휏 = 휏푛 − 훼푑푤|휎′ +푛|. +(44) +We set 훼푢푝 = 0.1, 훼푑푤 = 0.05, 푐 = 0 in Eq. (43), (44) during +simulations (see typical values of the friction angle and +cohesion in the fault zone in (Ikari et al., 2011; Treffeisen, +2021)). For estimation of the prefactor before Δ휏 in Eq. (40), +we assume that: (i) the fracture length 2푎 equals the height +of the mesh cell, which is close to 10 m, so that 푎 ∼ 5 m; (ii) +elastic modulus of the transversal isotropic body is given by +Bessmertnykh and Dontsov (2018): 퐶11 = 20 GPa, 퐶12 = +6.8 GPa, 퐶13 = 7.6 GPa, 퐶33 = 13 GPa, 퐶44 = 3 GPa; +(iii) the dilatancy coefficient is set to Λ = 0.5. As a result, +we obtain 퐸∗ +푑 ≅ 10−9Δ휏, where Δ휏 is in Pa. Note that the +dilatancy coefficient Λ varies in between 0 and 1 (Alejano +and Alonso, 2005). +Figure 7: Stress components 휏푛 and 휎푛 at the fault plane +inclined by angle 휓 to the vertical direction under plane stress +conditions. +If plastic deformations are observed, then the mechanical +fracture aperture 퐸푑 is calculated using dilatancy and shear +deformations as follows: +퐸푑 = Λ훾푑푥 +where 훾 is the intensity of shear deformations (second in- +variant of shear stress tensor), and 푑푥 is the mesh cell length +in direction perpendicular to the fault. +4. Results and discussion +4.1. Verification of coupled hydro-geomechanical +model +In the current section, we verify the implementation of +the in-house algorithm performing coupling between reser- +voir simulator MUFITS and mechanical simulator FLAC3D +via data transfer. Both simulators have been verified previ- +ously. Results of the benchmark tests of MUFITS are given +in papers (Afanasyev et al., 2016; De Lucia et al., 2016; +Afanasyev, 2017) and are available at the web-site of the +simulator (Afanasyev, 2020). At the same time, various tests +of the commercial simulator FLAC3D are provided in the +manual. +We consider the transient fluid flow to a vertical well +fully penetrating the infinite-acting aquifer with thickness ℎ. +We introduce the cylindrical coordinate system (푟, 휃, 푧) with +an origin located at the bottom of the perforation interval. +Fig. 8 shows the schematic representation of the model. +Figure 8: Schematic picture of a vertical production well fully +penetrating the infinite-acting aquifer. +Initially, the pore fluid pressure 푝0 is uniform, and the +reservoir is under isotropic stress 휎0 +푧푧. The vertical well +produces water at a constant rate 푞. The elastic porous +medium is taken as homogeneous and isotropic with drained +bulk modulus 퐾, shear modulus 퐺, Biot coefficient 훼, Biot +modulus 푀, porosity 휙, permeability 푘. Pore fluid (water) +is described by the viscosity 휇 and bulk modulus 퐾푓. We +consider an incompressible solid constituent 훼 = 1 so that +푀 = 퐾푓∕휙. The vertical stress is assumed constant during +the water production 휎푧푧(푟, 푡) = 휎0 +푧푧, and horizontal strains, +휀푟푟, 휀휃휃, are negligibly small as compared to 휀푧푧. +The solution to the formulated problem in terms of the +spatial-temporal parameters (pore fluid pressure 푝, stress +tensor components 휎푟푟, 휎휃휃, 휎푧푧, and vertical displacement +푢푧) can be derived analytically and is formulated as follows: +푝 = 푝0 − +푞휇 +4휋푘ℎ퐸1 +( +푟2 +4푐푡 +) +, +휎푟푟 = 휎휃휃 = 휎0 +푧푧 + 푞훼퐺휇 +2휋푘ℎ훼1 +퐸1 +( +푟2 +4푐푡 +) +, +휎푧푧 = 휎0 +푧푧, +Preprint submitted to Journal of Natural Gas Science and Engineering +Page 10 of 26 + +Z +山 +X +m +xzGeomechanical risks of CO2 storage +Parameter +Value, unit +퐾 +20 GPa +퐺 +10 GPa +퐾푓 +2 GPa +휇 +1 cP +휙 +0.1 +푘 +1 mD +훼 +1 +푞 +4 m3/day +푝0 +100 bar +휎0 +푧푧 +-100 bar +Table 1 +Parameters of the vertical well model considered in the +numerical experiments. +푢푧 = − 푧훼푞휇 +4휋푘ℎ훼1 +퐸1 +( +푟2 +4푐푡 +) +, +(45) +where 훼1 = 퐾 + 4퐺∕3, 푐 = 푘∕(휇푆) is the diffusion +coefficient, 푆 = 1∕푀 + 훼2∕훼1 is the storage coefficient. +We solve the formulated problem numerically using two +approaches: (i) developed coupled hydro-geomechanical +model based on MUFITS and FLAC3D and (ii) FLAC3D +(FLAC3D can simulate single phase flow of slightly com- +pressible liquid in addition to the mechanical calculations). +Subsequently, model (ii) is applied to verify the imple- +mentation of porosity and permeability alteration in the +hydrodynamical model according to Eq. (12), (13). +Table 1 outlines the values of model parameters used +in the simulations. Note that it is necessary to put the +adjusted fluid modulus into the hydrodynamic model 퐾푎 +푓 = +휙∕ (휙∕퐾푓 + 1∕훼1 +) in order to preserve the real diffusivity +(in the expression, we take into account the Biot coefficient +value 훼 = 1). We model the water production within 60 +days, and the outer boundary of the reservoir located in the +numerical models at a distance of 1 km from the producer +is not reached by the pressure wave during the simulation so +that the condition of the infinite-acting reservoir is satisfied. +Fig. 9 compares two numerical solutions computed by +MUFITS+FLAC3D and FLAC3D with the analytical so- +lution given by Eq. (45). We look at the distributions of +pressure, vertical displacement, radial and vertical stresses +over the coordinate interval 푟 ≤ 500 m at different time +moments. Since the vertical stress is constant over time, +we depict the numerical solution for this parameter at the +end of the simulation. One can conclude that the outcome +of the proposed coupled hydro-geomechanical model based +on MUFITS and FLAC3D matches acceptably with the +analytical solution of the problem, and we make similar +inference regarding the numerical model built on FLAC3D. +In the numerical experiment shown in Fig. 9, we check +the data transfer from MUFITS to FLAC3D. However, we +should also verify the reverse data flow between simulators +(i.e., from FLAC3D to MUFITS). For that purpose, we +amend the numerical models by adding the alteration of +porosity and permeability in the hydrodynamical part after +each mechanical calculation. We embed the following rela- +tions for porosity and permeability: 휙 = 1 − (1 − 휙0)푒−50⋅휀푣, +푘 = 푘0(휙∕휙0)8, where 휙0 = 0.1, 푘 = 1 mD, so that +we modify artificially Eq. (12) to increase the impact of +volumetric deformations on porosity, while the value of +the power-law exponent in Eq. (13) is within the typical +range. Moreover, in the current numerical experiment, it is +not required to adjust the fluid modulus. As a result, for +comparison purposes, we can demonstrate the offset of the +numerical solution from an analytical one as described by +Eq. (45), which corresponds to the storage coefficient 푆 = +휙∕퐾푓. +Fig. 10 shows the obtained results. We demonstrate the +distributions of pressure, vertical displacement, radial stress, +and permeability over the coordinate interval 푟 ≤ 100 m +at different time instants (permeability evolution is shown +within the area 푟 ∈ [0, 500] m). The numerical solutions +deviate from the analytical ones after a couple of days of +water production, the discrepancy is observed at 푡 = 5 days. +We would like to stress that the analytical curves in Fig. 10 +are not the solution to the problem under consideration, +and they correspond to the water flow in an incompressible +porous medium with constant porosity 휙0 = 0.1 and perme- +ability 푘0 = 1 mD. Moreover, one can observe a satisfac- +tory match between the results of simulations obtained via +MUFITS+FLAC3D and FLAC3D. In other words, Fig. 10 +confirms the correct implementation of the data transfer +from FLAC3D to MUFITS performed via the porosity and +permeability modification in the hydrodynamical model im- +plemented in MUFITS based on deformations computed by +FLAC3D. +4.2. Coupled hydro-geomechanical modeling of +CO2 sequestration in an aquifer on the +example of the synthetic formation case +The current section presents the simulation results of +CO2 injection into the target aquifer intersected by the +tectonic fault. We construct a synthetic model of two- +dimensional multilayered formation. During the interpre- +tation of the calculations, we focus on the development of +undesired mechanical processes, namely, slip along the fault +plane resulting in the crack opening on the asperities, the +plastic deformations in the fault zone, target aquifer, and +caprock, as well as carbon dioxide leakage along the fault +zone towards the overlying collector. +4.2.1. Model description +Fig. 11 shows the schematic representation of the utilized +synthetic model of a two-dimensional multilayered reservoir +with the following geometrical parameters: the lateral size of +the reservoir is 1500 m, while its height equals 600 m. Two +cases of the reservoir depth are considered: the upper bound +locates at a depth of (i) 600 m and (ii) 1700 m. +Carbon dioxide is injected into the target aquifer of +thickness 100 m surrounded by the low permeable caprock +and basement layers, each of 150 m height. These three +layers are intersected by the tectonic fault. The fault zone +Preprint submitted to Journal of Natural Gas Science and Engineering +Page 11 of 26 + +Geomechanical risks of CO2 storage +Figure 9: Results of the analytical and numerical modeling of transient fluid flow to a vertical well fully penetrating the infinite- +acting poroelastic reservoir in terms of pressure (a), vertical displacement (b), radial stress (c) and vertical stress (d) distributions; +The analytical solution is shown by solid lines, while the numerical solutions are given by markers, MUFITS+FLAC3D by crosses +and FLAC3D by circles; the solutions correspond to the following time instants: 푡 = {1, 5, 15, 30, 60} day(s); zoomed domain in +the vicinity of the well 푟 ∈ [0, 100] m is shown in plots (a) – (c). +thickness is 20 m, the inclination angle relative to the vertical +direction is 15◦. The distance between the left edge of the +formation and the fault at a depth corresponding to the +middle of the storage aquifer is 550 m. The fault consists +of the damage zone and core, the thickness of latter one is 1 +m. The upper and basal aquifers are placed above and below +the caprock and basement layers. The injector has a vertical +completion coinciding with the left border of the reservoir. +CO2 injection is carried out through the perforations located +along the intervals: (i) 푧 ∈ [−910, −890] m and (ii) 푧 ∈ +[−1910, −1890] m. +Before CO2 injection, the formation is assumed to be +water-saturated, the pressure distribution is hydrostatic (in +the general case, it is computed from the phases distribution +and gravitational-capillary equilibrium), and the tempera- +ture field corresponds to the geothermal gradient of 25 +◦C/km. FLAC3D computes the in-situ mechanical state of +the reservoir using the prescribed pressure and temperature +at the initial time instant and the geological model (distri- +butions of density and elasticity modulus). The calculated +initial displacements are set to zero, so that the subsequent +deformations appearing due to CO2 injection are measured +from the in-situ state. The vertical well injects carbon diox- +ide at constant bottomhole pressure: (i) 150 bar and (ii) 300 +bar. We choose the bottomhole pressure values relying on +the minimal principal stresses 휎min observed at a depth of +perforations in the first (i) and second (ii) cases as follows: +bottomhole pressure should be lower than 휎min (e.g., by +10 %), to prevent the initiation of hydraulic fractures. The +top and left edges of the reservoir are impermeable (solid +black lines in Fig. 11). At the bottom and right boundaries, +we specify a constant pore pressure equal to the initial one +(dashed black lines in Fig. 11). At the left and right edges of +the formation, we fix zero displacements along x-axis 푢푥 = +0, while the displacements along x and z-axis are prohibited +at the bottom boundary 푢푥 = 푢푧 = 0. In addition, we fix the +displacement along z-axis at the right border 푢푧 = 0. At the +Preprint submitted to Journal of Natural Gas Science and Engineering +Page 12 of 26 + +a) 100 +c) +-75 +75 +-80- +-80 - +90- +100- +85 +Orr, bar +-85- +-90- +bar +90 +bar +80 +Orr, +-95- +80 +-06- +100 +0 +20 +40 +60 +80 +100 +70- +70 +区 +r, m +-95 +X +60 +0 +20 +40 +60 +80 +100 +-100 +60- +r, m +0 +100 +200 +300 +400 +500 +0 +100 +200 +300 +400 +500 +r, m +r, m +b) +0.0 +-95 +analytical +t= +-96- +X +MUFITS+FLAC3D +1 d +-0.2 +0 +FLAC3D +5 d +X +0.0 +-97 +15 d +ww +-0.4 +0.2 +bar +-98- +30 d +Ozz' +60 d +-0.6 +-0.6- +-99 +-0.8- +0.8 +X +-100 +-1.0 +0 +20 +40 +60 +80 +100 +1.0 +r, m +-101 +0 +100 +200 +300 +400 +500 +0 +100 +200 +300 +400 +500 +r, m +r, mGeomechanical risks of CO2 storage +Figure 10: +Results of the numerical modeling of transient axisymmetric fluid flow to a vertical well fully penetrating the +infinite-acting reservoir accounting for the porosity and permeability alterations based on the mechanical calculations: pressure +(a), vertical displacement (b), radial stress (c) and permeability (d) distributions; The numerical solutions are given by markers, +MUFITS+FLAC3D by crosses and FLAC3D by circles; analytical solution corresponding to the pore fluid flow in the incompressible +porous medium with the constant porosity and permeability (dashed lines); we demonstrate the solutions along the spatial domain +푟 ∈ [0, 100] m (푟 ∈ [0, 500] m in plot d) at the following time instants: 푡 = {1, 5, 15, 30, 60} day(s). +Figure 11: Synthetic reservoir model; solid and dashed black +lines denote impermeable and constant pressure boundaries, +respectively; the red arrow marks the perforation interval +through which CO2 is injected into the target aquifer; by red +and purple colors we show core and damage zones in the fault +domain. +top boundary, we apply constant loading 휎푧푧 corresponding +to the lithostatic pressure created by a layer of thickness (i) +600 m and (ii) 1700 m with a density 2400 kg/m3. +Table 2 provides the model parameters: +• mechanical properties: Young’s modulus 퐸, Poisson +ratio 휈, cohesion 푐, angle of internal friction 휃, dila- +tancy Λ; +• porosity 휙, permeability 푘; +• rock density 휌. +We set the specific heat capacity of rock 퐶푟 = 0.81 kJ / +(kg ⋅ K) and heat conductivity of saturated porous medium +휆 = 3 W / (m ⋅ K) for the entire domain. Pore water salinity +is neglected. +Note that the chosen permeability values of the dam- +age zone and fault core are consistent with the experi- +mental measurements and estimations provided in papers +Preprint submitted to Journal of Natural Gas Science and Engineering +Page 13 of 26 + +a) 100- +C +-70 +t = +-75- +1 d +X4 +90 +5 d +区 +-80 +15 d +bar +bar +X +80 +30 d +X +-85 +Orr, +X +p +60 d +X +-90 +由 +X +70 +区 +analytical Kf +X +区 +这 +X +-95 +区 +X +MUFITS+FLAC3D +60 +8 +0 +FLAC3D +-100 +0 +20 +40 +60 +80 +100 +0 +20 +40 +60 +80 +100 +r, m +r, m +b) +0.0. +d) +1.00 +& +& +α +& +区 +& +X +& +-0.2 +0.95 +区 +区 +& +& +R +X +区 +& +& +0.90 +& +-0.4 +文 +& +X +ww +0.85 +D +m +& +& +-0.6 +zn +K 0.80 +& +文 +-0.8 - +& +0.75 +-1.0- +0.70 +X +0.65 8 +-1.2 +0 +20 +40 +60 +80 +100 +0 +100 +200 +300 +400 +500 +r, m +r, mCO2 +X +Upper aquifer +100 m +Caprock +150 m +550 m +Target aquifer +100 m +:15° +Basement +150 m +20 m +Basal aguifer +100 m +1500 mGeomechanical risks of CO2 storage +Layer +휙 +푘, mD +퐸, GPa +휈 +휌, kg/m3 +푐, MPa +휃 +Λ +Upper aquifer +0.1 +10 +20 +0.2 +2400 +- +- +- +Caprock +0.01 +10−4 +20 +0.15 +2400 +4 +24 +0.2 +Target aquifer +0.1 +100 +10 +0.2 +2400 +5.6 +40 +0.4 +Basement +0.01 +10−4 +20 +0.3 +2600 +- +- +- +Basal aquifer +0.01 +0.1 +20 +0.3 +2600 +- +- +- +Fault (damage zone) +0.1 +0.1 +- +- +2400 +- +- +- +Fault (core) +0.1 +10−3 +- +- +2400 +- +- +- +Table 2 +Mechanical and flow properties of the synthetic reservoir model depicted in Fig. 11. The values of cohesion, angle of internal +friction, and dilatancy corresponding to the upper aquifer, basement, and basal aquifer are absent, since these layers are considered +as elastic. We set the values of the elastic modulus and strength parameters in the fault zone in such a way as to reproduce its +complex structure, and one can find the description of this procedure in the main text. +(Faulkner et al., 2003; Wibberley and Shimamoto, 2003; +Scibek, 2020). The authors of these studies conclude that +the fault core permeability is typically lower than that of the +damage zone. As a result, the fault permeability along its +plane is larger than the permeability in the normal direction. +Fig. 12 shows the utilized relative permeabilities for +the gas (black line) and liquid (red line) phases, as well as +the capillary pressure curve (blue line). These parameters +depend on the liquid phase saturation 푠푙 according to Eqs. (7) +with 푠푙푟 = 0.3, 푠푔푟 = 0.05, 휆푙 = 휆푐 = 0.457, 푃푐0 = 0.1961 +bar. +Figure 12: Relative permeabilities and capillary pressure curves +embedded into the hydrodynamic reservoir model. +We specify that the target aquifer and caprock including +the intersected fault zone are described by the elastoplas- +tic rheological model based on the Drucker-Prager yield +condition. The remaining layers are elastic. The choice is +motivated by the aim to track the development of plastic +deformations in the regions ensuring the loss of integrity of +the carbon dioxide storage. +The spatial meshes in the reservoir simulator MUFITS +and mechanical simulator FLAC3D are identical. The di- +mensions of mesh cells out of the fault zone are Δ푥 = 20 +m, Δ푧 = 10 m (Fig. 13) with slightly variation towards the +fault, where the columns of cells become parallel to the fault +plane. We apply the grid refinement in the fault zone, which +allows reproducing its complex structure including the core +surrounded by the damage zone. Seven columns of cells are +used to approximated the fault zone, and their thickness de- +creases towards the fault center. The central column denotes +the fault core and contains the main crack, which is initially +healed. The remaining 6 columns (3 to the left and right of +the fault core) describe the damage zone. Elastic modules +(bulk modulus 퐾 and shear modulus 퐺), cohesion 푐, and +angle of internal friction 휃 decrease towards the fault core in +accordance with the studies Gudmundsson (2004); Faulkner +et al. (2006); Treffeisen (2021). We assume that the values +of these parameters at the fault core comprise 70% of those +corresponding of the host rock at the considered depth (see +typical values of the contrast in the mechanical properties +between the host rock and fault core in Holdsworth (2004); +Collettini et al. (2009); Treffeisen (2021). The trend for +variation of the dilatancy coefficient Λ is similar, but it +increases towards the fault center so that its values at the +fault core are 30% larger than that in the host rock. +Figure 13: Reservoir spatial discretization; the mesh structure +in the fault zone is zoomed; colors highlight different layers +and fault zone (see Table 2). +CO2 injection is simulated for 30 years. During the first 5 +years, we compute the mechanical equilibrium state every 6 +months using FLAC3D simulator and reservoir porosity and +permeability are updated according to the current stress state +and deformations. After that period, FLAC3D simulator is +called once a year for 25 years. +4.2.2. Modeling results +Target aquifer at 950m depth +We start with discussion of the results of pure hydrody- +namical modeling of CO2 injection using MUFITS simula- +tor (see Fig. 14). After 30 years of injection, the CO2 plume +Preprint submitted to Journal of Natural Gas Science and Engineering +Page 14 of 26 + +.6 +0.95 +KRGAS +1.5 +0.9. +KRLIQ +1.4 +0.85 +PCGL +0.8 +1.3 +0.75 +1.2 +es +0.7. +bar +1.1 +0.65 +igpi +pressure, +0.6 +e +0.9 +0.55 +0.5 +0.8 +.0.45 +0.7 +Capillary +0.4 +0.6 +0.35 +0.5 +Rel +0.3 +0.25 +0.4 +0.2 +0.3 +0.15 +0.2 +0.1. +0.05 +0.1 +0. +0 +0 +0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 +Liquid saturation-600 +-650 +-700 +-750 +-800 +-850 +-900 +N.950 +-1000 +-1050 +-1100 +-1150 +-1200 +-500 -400 -300 -200 -100 +0 +100 +200300 +400 +500 +600 +700 +800 +9001000 +x,mGeomechanical risks of CO2 storage +reaches the fault zone and crosses it (see Fig. 14a). On the +right side of the fault, CO2 flows along the upper part of +the target aquifer and passes 400 m. Carbon dioxide also +flows along the tectonic fault where the CO2 plume uplifts +at a distance of 50 m. From Fig. 14b, one can notice that +the storage aquifer leftward to the fault zone contains the +pressure plume. Increase in pore pressure (the difference in +pore pressure values at the final and initial time instants) is +homogeneous in the target aquifer due to the high contrast +in permeability values corresponding to the CO2 storage do- +main and surrounding layers, where the pressure disturbance +gradually decreases. On the left hand of the fault, the pore +pressure increase is observed in the caprock and basement +layers only. Consequently, according to the hydrodynamic +simulation, there is no leakage of CO2 from the target layer +to the upper aquifer. +Figure 14: +Results of hydrodynamical modeling of CO2 +injection into the target aquifer with the upper boundary +located at a depth of 600 m; plots (a) and (b) show the +gas saturation and pore pressure increment (as compared to +the initial hydrostatic distribution) fields at the end of the +simulation period (30 years). +In the framework of coupled simulations we consider +two cases of the initial mechanical state: 휎푥푥 = 휎푦푦 (no +tectonic stresses) and 휎푦푦∕휎푥푥 ∼ 2.7 at the depth of the target +aquifer. +We begin with the case of no tectonic stresses, and +Fig. 15 presents the results of simulations. Carbon dioxide +flows along the fault, and the CO2 plume reaches the upper +aquifer as it can be noted in Fig. 15a. The opening of the +main crack facilitates this behavior. During CO2 injection, +the slip along the fault plane in an elastic mode occurs, and +the fracture opens at natural asperities. After 30 years of +CO2 sequestration, the main crack opens along its entire +length. However, the fracture aperture at the depth interval +corresponding to the caprock and storage aquifer is smaller +as compared to that at the basement layer. Plastic deforma- +tions do not develop in the reservoir so that the permeability +increase in the fault zone is observed only in the direction +parallel to the fault plane due to opening of the main fracture. +The appearance of the conduit leads to CO2 flow along the +fault to the upper aquifer and the carbon dioxide leakage out +of the disposal zone. Moreover, rightward to the fault zone, +CO2 flow occurs along the shorter distance compared to that +obtained using pure hydrodynamical modeling (Fig. 14a). +Fig. 15d demonstrates the mass of the injected CO2 in +the coupled model (red line) and hydrodynamical model +(blue line), and the former one is higher. The pore pressure +increment shown in Fig. 15b is similar to that obtained using +hydrodynamical model (Fig. 14b). However, the pressure +increase on the right side of the fault in the caprock and +basement layers is more pronounced in the coupled model. +Volumetric strain is maximal in the target aquifer leftward to +the fault and declines towards the upper and basal aquifers +in the caprock and basement layers (see Fig. 15c). +Next, we discuss the results of the coupled hydro- +geomechanical modeling of CO2 injection into the target +aquifer with pronounced tectonic stresses at the initial state +as shown in Fig. 16. The ratio 휎푦푦∕휎푥푥 ∼ 2.7 is set to +observe the development of plastic deformations in the fault +zone. After 30 years of carbon dioxide injection, the main +crack opens along the entire length, and the fracture aper- +ture decreases towards the basement layer. Intense plastic +deformations develop in the fault zone at the depth of the +target aquifer so that the natural fractures open in the damage +zone of the tectonic fault. Thereby, the permeabilities of the +fault along and perpendicular to its plane are increased. As a +result, we observe a significant CO2 leakage into the upper +aquifer and CO2 flow in the target aquifer on the right side +of the fault (see Fig. 16a). From Fig. 16d, it is clear that the +injected mass of CO2 in the coupled hydro-geomechanical +model is much higher as compared to the hydrodynamic +simulation. The distributions of the pore pressure increment +and volumetric strain (Figs. 16b and c) are similar to that +obtained in the case of no tectonic stresses (Fig. 15) except +for the domain of the upper aquifer where both parameters +grow due to the leakage of carbon dioxide. Additionally, note +the significant volumetric strain and pore pressure in the fault +zone at a depth of the caprock layer. +Target aquifer at 2000m depth +Now we discuss the results of the modeling of CO2 +injection into the target aquifer of the formation with the +upper boundary located at the depth of 1700 m. We begin +with the hydrodynamic simulation (see Fig. 17). At the end +of the simulation period (30 years), the CO2 plume reaches +the fault zone and crosses it (see Fig. 17a). During the +flow rightward to the fault, CO2 passes 900 m reaching +approximately the right boundary of the reservoir. We also +observe CO2 flow along the fault zone, and carbon dioxide +rises at a distance of 150 m. The shape of the pore pressure +plume demonstrated in Fig. 17b is similar to that obtained in +the previous case of the reservoir depth of 950 m (Fig. 14b). +The difference is that the pressure increase with respect to +the initial distribution is higher due to larger bottomhole +pressure value (300 bar versus 150 bar). Thus, the hydrody- +namic computation demonstrates that the CO2 plume almost +reaches the upper aquifer. +Preprint submitted to Journal of Natural Gas Science and Engineering +Page 15 of 26 + +a) +-600 +-650 +-700 +-750 +7.0e-01 +-800 +0.6 +0.5 +0.4 +N-950 +0.3 +-1000 +0.2 +-1050 +-1100 +0.1 +-1150 +0.0e+00 +-1200 +-500 +-400-300-200-100 +0 +100 +200300 +400 +500 +600 +700 +800 +9001000 +x,m +b) +-600 +-650 +Ap, bar +-700 +-750 +6.5e+01 +-800 +50 +40 +N-950 +30 +-1000 +20 +-1050 +-1100 +10 +-1150 +0.0e+00 +-120Q +-500 +-400-300 +-200 +-100 +0 +100 +200300 +400 +500 +600 +700 +800 +9001000 +x,mGeomechanical risks of CO2 storage +Figure 15: Results of coupled hydro-geomechanical modeling of CO2 injection into the target aquifer with the upper boundary +located at a depth of 600 m in the absence of tectonic stresses (휎푥푥 = 휎푦푦); plots (a) – (c) show the gas saturation, pore pressure +increment (as compared to the initial hydrostatic distribution), and volumetric strain fields at the end of the simulation period +(30 years), respectively; plot (d) depicts the dynamics of injected mass of carbon dioxide in pure hydrodynamical model (red +curve) and coupled model (blue curve). +Figure 16: Results of coupled hydro-geomechanical modeling of CO2 injection into the target aquifer with the upper boundary +located at a depth of 600 m at the pronounced tectonic stresses (휎푦푦∕휎푥푥 ∼ 2.7 in the target aquifer); plots (a) – (c) show +the gas saturation, pore pressure increment (compared to the initial hydrostatic distribution), and volumetric strain fields at the +end of the simulation period (30 years), respectively; plot (d) depicts the dynamics of injected mass of carbon dioxide in pure +hydrodynamical model (red curve) and coupled model (blue curve). +Let us consider the results of the coupled simulations +in the absence of tectonic stress at the initial mechanical +state 휎푥푥 = 휎푦푦 (see Fig. 18). Plastic deformations do not +develop in the reservoir, and the main crack opens along +the entire length in the elastic mode. The formed conduit +contributes to the leakage of carbon dioxide into the upper +aquifer. The CO2 plume crosses the fault zone and spreads +along the target layer on the right side of the fault over a +shorter distance as compared to that obtained using pure +hydrodynamical calculations (Fig. 18a). Fig. 18d compares +the mass of injected CO2 in the coupled and hydrodynamical +simulations, and the former one is higher due to the major +crack opening on asperities. The shape of the pressure plume +shown in Fig. 18b is similar to that obtained in the previous +configuration of the formation with no tectonic stresses +(Fig. 15b). The target aquifer exhibits large values of the +volumetric strain (Fig. 18c). We also observe the reduced +volumetric deformations in the vicinity of the injector. We +attribute this to the temperature effect since the injected +carbon dioxide is colder as compared to the reservoir water +inside the target aquifer. In the present case, CO2 cools the +reservoir in the vicinity of injector by 40 degrees, while in +the previous case (Fig. 15c), we do not observe noticeable +influence of the non-isothermal flow on the mechanical +equilibrium state of the formation due to small difference +in between CO2 and pore fluid temperatures. +Preprint submitted to Journal of Natural Gas Science and Engineering +Page 16 of 26 + +a) +-600 +c) +-600 +-650 +-650 +& +-700 +-700 +-750 +7.0e-01 +-750 +7.0e-04 +-800 +0.6 +-800 +0.0006 +0.5 +0.0005 +0.4 +0.0004 +Zi-950 +Z-950 +0.3 +0.0003 +-1000 +-1000 +0.0002 +-1050 + 0.2 +-1050 +0.0001 +-1100 +0.1 +-1100 +-1150 +0.0e+00 +-1150 +5.0e-05 +-120Q +-1200 +1002003004005006007008009001000 +0 +100200 +3004005006007008009001000 +x,m +x,m +-600 +9.5e+3 +b) +-650 +Ap, bar +d) +8.5e+3 +COUPLED +-700 +8.0e+3 +-750 +6.5e+01 +7.5e+3 +7.0e+3 +-800 +6.5e+3 +50 +40 +N.950 +4.5e+3 +.0e+3 +30 +-1000 +4.0e+3 +20 +3.5e+3 +-1050 +3.0e+3 +-1100 +10 +2.5e+3 +2.0e+3 +-1150 +0.0e+00 +1.5e+3 +120Q +1.0e+3 +500-400-300-200-1000 +1002003004005006007008009001000 +5.0e+2 +x,m-600 +c) +-600 +a) +-650 +-650 +-700 +-700 +-750 +7.0e-01 +-750 +7.0e-04 +-800 +0.6 +-800 +0.0006 +0.5 +≤900 +-850 +0.0005 +0.4 +0.0004 +N-950 +N-950 +0.3 +0.0003 +-1000 +0001- +0.2 +-1050 +0.0002 +-1050 +-1100 +0.1 +-1100 +0.0001 +-1150 +0.0e+00 +-1150 +5.0e-05 +-120Q +-120Q +500-400-300-200-1000 +0001 006008002009009000000002001 +-500-400-300-200-1000 +1002003004005006007008009001000 +x,m +x,m +-600 +b) +-650 +Ap, bar +d) +18814 +-COUPLED +-700 +-750 +6.5e+01 +1.7e+4 +-800 + 50 +.4e+4 +.3e+4 +-900 +40 +Z-950 +30 +-1000 +E9.0e +20 +3.0e+3 +-1050 +-1100 +10 +-1150 +0.0e+00 +4.0e+3 +3.0e+3 +-1200 +-500-400-300-200-1000 +1002003004005006007008009001000 +x,mGeomechanical risks of CO2 storage +Figure 17: +Results of hydrodynamical modeling of CO2 +injection into the target aquifer with the upper boundary +located at the depth of 1700 m; plots (a) and (b) show the gas +saturation and pore pressure increment (compared to the initial +hydrostatic distribution) fields at the end of the simulation +period (30 years), respectively. +Fig. 19 shows the results of coupled hydro-geomechanical +modeling of CO2 injection into the formation in the presence +of pronounced tectonic stresses. The ratio of principal +stresses at the target layer 휎푦푦∕휎푥푥 ∼ 1.8 is set to facilitate +the development of plastic deformations in the fault zone. +Note the plastic deformations are formed at a lower value +of the ratio 휎푦푦∕휎푥푥 as compared to the prevoius case of +target aquifer located at the depth of 950 m. After 30 years of +carbon dioxide injection, the main fracture opens along the +entire length of the fault zone. We obtained smaller values of +the crack aperture at the depth interval corresponding to the +bottom segment of the storage aquifer. Moreover, we observe +the development of the plastic deformations at the fault zone +at the caprock layer and target aquifer. The opened major +crack in the core and natural fractures in the damage zone of +the fault contribute to the CO2 leakage into the upper aquifer +and CO2 flow towards the right border of the reservoir (see +Fig. 19a). The latter effect results in the larger volume of CO2 +plume rightward to the fault as compared to that obtained +using the hydrodynamic model. The mass of injected CO2 +is substantially larger in the case of the coupled model (red +curve in Fig. 19d) as compared to that obtained using the +hydrodynamic model. The pore pressure in the plume differs +from that obtained in the case of no tectonic stresses by a +tangible pressure increase in the upper aquifer due to the +carbon dioxide leakage (Fig. 19b). Volumetric deformation +is maximal in the fault zone at the depth of the caprock layer +(Fig. 19c). Still they are large in the target aquifer on the left +side of the fault. The reduced values of the volumetric strain +are attributed to the thermal effects. +4.3. Coupled hydro-geomechanical modeling of +CO2 sequestration in an aquifer of the real +formation +In the current section, we show the results the modeling +of CO2 injection into an aquifer using the proposed coupled +hydro-geomechanical approach. We consider a slice of the +reservoir sector that contains a tectonic fault. Similar to +Section 4.2, the model is two-dimensional. For the model +construction we use the field data collected from well log- +ging, well test, seismic survey, and laboratory experiments. +4.3.1. Model description +Fig. 20 shows the schematic representation of the for- +mation under consideration. Here, we illustrate its structure, +geometrical characteristics, and the tectonic fault placement +relative to the injector. +The reservoir has the length and height of 1500 m and +1350 m, respectively. The upper boundary of the formation +is located at the depth of 1350 m. Carbon dioxide is injected +into the upper and lower aquifers through a vertical well +located at the left boundary of the reservoir; perforations +are distributed along the interval 푧 ∈ [−2300, −2100] m +except for the 10 m thick caprock layer. From Fig. 20 one can +observe a layer of anhydrite and a massive layer of salt above +the upper aquifer. A tectonic fault starts at the salt layer and +finishes at the basement crossing the anhydrite layer as well +as aquifers and caprock. The fault is almost vertical with an +inclination angle of 3◦ with respect to the vertical direction. +The distance between the injector and the fault equals 360 +m at 푧 = −2300 m corresponding to the middle of the lower +aquifer. We embed the same fault structure into the coupled +model as considered in Section 4.2, namely, the fault has +thickness of 20 m, while its core thickness is set to 1 m. +Carbon dioxide is injected into the upper and lower +aquifers at fixed bottomhole pressure. We consider two cases +of injection with the following bottomhole pressures: (i) 350 +bar (base case) and (ii) 600 bar (upper limit). The boundary +conditions in the hydrodynamic and mechanical models are +similar to the synthetic reservoir model. The difference is +in the constant loading applied at the upper border of the +formation: in the current model, 휎푧푧 corresponds to the +lithostatic pressure created by a layer of thickness 1350 m +with a density 2400 kg/m3. +We describe the mechanical and flow properties of the +formation in Table 3. The relative permeabilities and capil- +lary pressure curve are the same as provided in Fig. 12. +Principal components of the tectonic strain tensor at the +initial state are 휀푡 +푥푥 = −10−4, 휀푡 +푦푦 = −3 ⋅ 10−4 at a depth of +the lower aquifer. Using the relations +Σ푡 +푥푥 = +퐸 +(1 − 휈2) +( +휀푡 +푥푥+휈휀푡 +푦푦 +) +, Σ푡 +푦푦 = +퐸 +(1 − 휈2) +( +휀푡 +푦푦+휈휀푡 +푥푥 +) +, +we compute the principal components of the tectonic stress +tensor. In these relations, Young’s modulus 퐸 and Poisson +ratio correspond to the lower aquifer. Since we are interested +in the tectonic stresses observed at the plane of the examined +Preprint submitted to Journal of Natural Gas Science and Engineering +Page 17 of 26 + +-1700 +a) +-1750 +S +-1800 +gas +-1850 +7.0e-01 +-1900 +0.6 +0.5 +0.4 +N-2050 +0.3 +-2100 +-2150 +0.2 +-2200 +0.1 +-2250 +0.0e+00 +-2300 +-500 +-400-300-200-100 +0 +100 +200 +300 +400 +500 +600 +700 +8009001000 +x,m +-1700 +b) +-1750 +Ap, bar +-1800 +-1850 +1.1e+02 +-1900 +80 +N-2050 +60 +-2100 +40 +-2150 +20 +-2200 +-2250 +0.0e+00 +-230Q +-500 +-400-300 +-200 +-100 +0 +100 +200300 +400 +500 +600 +700 +800 +9001000 +x,mGeomechanical risks of CO2 storage +Figure 18: Results of coupled hydro-geomechanical modeling of CO2 injection into the target aquifer with the upper boundary +located at a depth of 1700 m with no tectonic stresses (휎푥푥 = 휎푦푦); plots (a) – (c) show the gas saturation, pore pressure +increment (compared to the initial hydrostatic distribution), and volumetric strain fields at the end of the simulation period (30 +years), respectively; plot (d) depicts the dynamics of injected mass of carbon dioxide in pure hydrodynamical model (red curve) +and coupled model (blue curve). +Figure 19: The figure presents the results of coupled hydro-geomechanical modeling of CO2 injection into the target aquifer with +the upper boundary located at a depth of 1700 m accounting for the tectonic stresses (휎푦푦∕휎푥푥 ∼ 1.8 in the target aquifer). Panels +(a) – (c) show the gas saturation, pore pressure increment (compared to the initial hydrostatic distribution), and volumetric strain +fields at the end of the simulation period (30 years). Panel (d) depicts the dependence of the injected mass of carbon dioxide on +time. +Layer +휙 +푘, mD +퐸, GPa +휈 +휌, kg/m3 +푐, MPa +휃 +Λ +Salt +0.01 +10−4 +8 +0.44 +2100 +4.5 +28 +0.1 +Anhydrite +0.05 +0.05 +40 +0.26 +3000 +16.1 +35 +0.1 +Upper aquifer +0.14 +0.4 +36 +0.3 +2600 +12.4 +39 +0.1 +Caprock +0.01 +10−4 +38 +0.3 +2680 +17.1 +29 +0.1 +Lower aquifer +0.15 +0.6 +34 +0.27 +2610 +13.4 +38 +0.1 +Basement +0.01 +10−4 +38 +0.3 +2680 +17.1 +29 +0.1 +Fault (damage zone) +0.1 +0.6 +- +- +2600 +- +- +- +Fault (core) +0.1 +10−2 +- +- +2600 +- +- +- +Table 3 +Mechanical and flow properties of the realistic formation shown in Fig. 20); all layers are governed by the elastoplastic constitutive +model; the distribution of the mechanical properties inside the fault zone is similar to that described in Section 4.2. +Preprint submitted to Journal of Natural Gas Science and Engineering +Page 18 of 26 + +-1700 +-1700 +a) +-1750 +c) +-1750 +S +Ev +-1800 +gas +-1800 +-1850 +7.0e-01 +-1850 +1.0e-03 +-1900 +0.6 +-1900 +0.0008 +0.5 +1-2000 +-2000 +0.0006 +N.-2050 +0.4 +N-2050 +0.3 +0.0004 +-2100 +-2100 +-2150 + 0.2 +-2150 +0.0002 +-2200 +- 0.1 +-2200 +-2250 +0.0e+00 +2250 +7.0e-05 +-2300 +230Q +500-400-300-200-1000 +1002003004005006007008009001000 +500-400-300-200-100 +100 +x,m +x,m +-1700 +2.8e+4 +b) +-1750 +Ap, bar +d) +2.6e+4 +COUPLED +-1800 +2.4e+4 +-1850 +1.1e+02 +2.2e+4 +-1900 +2.0e+4 +80 +1.8e+4 +1.6e+4 +60 +1 +N-2050 +-2100 +- 40 +1.0e+4 +-2150 +- 20 +8.0e+3 +-2200 +6.0e+3 +-2250 +0.0e+00 +4.0e+3 +2300 +500-400-300-200-1000 +1002003004005006007008009001000 +2.0e+3 +x,m-1700 +-1700 +a) +-1750 +c) +-1750 +Ev +-1800 +-1800 +-1850 +7.0e-01 +-1850 +1.0e-03 +-1900 +0.6 +-1900 +0.0008 +0.5 +0.0006 +N-2050 +0.4 +Ni-2050 +-2100 +0.3 +0.0004 +-2100 +-2150 + 0.2 +-2150 +0.0002 +-2200 +0.1 +-2200 +-2250 +0.0e+00 +-2250 +-7.0e-05 +-230Q +500-400-300-200-1000 +1002003004005006007008009001000 +2300 +500-400-300-200-1000 +1002003004005006007008009001000 +x,m +x,m +b) +1700 +-1750 +Ap, bar +d) +-1800 +1.3e+5 +1850 +1.1e+02 +1.2e+5 +1.1e+5 +-1900 +1.0e+5 +80 +8.0e+4 +9.0e+4 +Ni-2050 +60 +7.00+4 +-2100 +40 + 6.0e+4 +2150 +5.0e+4 +20 +-2200 +4.0e+4. +0.0e+00 +3.0e+4 +-2250 +2.0e+4 +2300 +1002003004005006007008009001000 +1.0e+4 +500-400-300-200-1000 +x,m +0.0 +23456789101112131415161718192021222324252627282930Geomechanical risks of CO2 storage +Figure 20: +The slice of the sector of the real formation; by +solid and dashed black lines we show impermeable and constant +pressure boundaries, respectively; the yellow bars mark the +perforation intervals through which CO2 is injected into the +upper and lower aquifers. +reservoir slice, we utilize the following expressions: +휎푡 +푥푥 = Σ푡 +푥푥 cos2 휓 + Σ푡 +푦푦 sin2 휓, +휎푡 +푦푦 = Σ푡 +푥푥 sin2 휓 + Σ푡 +푦푦 cos2 휓, +휎푡 +푥푦 = 1 +2 +( +Σ푡 +푦푦 − Σ푡 +푥푥 +) +sin 2휓, +where 휓 is the angle between the principal x-direction and +the slice plane (see Fig. 21). In the current case, the angle 휓 +equals 15◦. +Figure 21: +Components of the tectonic stress tensor in the +coordinate axes 푥′푦′ (휎푡 +푖푗) and in the principal axes 푥푦 (Σ푡 +푖푗). +In the mechanical reservoir model, we describe the +rheology of all layers by the elastoplastic model with the +Drucker-Prager yield condition. The spatial meshes in MU- +FITS and FLAC3D are identical. The cell dimensions are +Δ푥 = 20 m and Δ푧 = 10 m in the domain outside of the +fault zone, where we utilize the same grid refinement in the +fault zone as in the synthetic reservoir model (Section 4.2) to +represent its complex structure. Carbon dioxide is injected +for 30 years. +4.3.2. Modeling results +We start with the results of the base case. Here, carbon +dioxide is injected into the upper and lower aquifers at a +constant bottomhole pressure of 350 bar. Fig. 22 shows +the gas saturation, pore pressure increment, and volumetric +strain distributions at the end of the simulation period. +After 30 years of CO2 injection, the carbon dioxide +plume does not reach the fault zone (Fig. 22a). Its maximum +lateral size and height are 250 m and 450 m, respectively. +Large pore pressure increase is observed inside the domain +occupied by the CO2 plume (Fig. 22b). The pore pressure +perturbations extend across the entire thickness of the reser- +voir and along the distance of 1 km from the injector in the +lateral direction. Thus, the pore pressure plume dimensions +are much larger as compared to that of CO2 plume. We +observe the large values of the volumetric strain in both +aquifers and in the lower part of the salt deposit (Fig. 22c). +Similar to the synthetic reservoir model (Section 4.2), the +reduced values of the volumetric strain near the perforations +are attributed to the cooling effect. Plastic deformations are +not developed in the reservoir. The main fracture does not +open so that the condition Eq. (43) is not satisfied along the +entire crack. +In the current case, permeability can increase in the for- +mation due to the volumetric deformations only according +to Eq. (13). Since deformations are relatively small, the +changes in porosity and permeability are also small. For +example, permeability increase is less than 1%. Comparing +dynamics of the mass of injected carbon dioxide in the +case of coupled modeling and hydrodynamic simulation, we +obtain a negligible difference between them. +Next, we move on to the results of the case in which the +bottomhole pressure is fixed to 600 bar, which exceeds the +minimum principal stress. Fig. 23 illustrates the distributions +of the gas saturation, pore pressure increment, and volumet- +ric strain after 30 years of CO2 injection. +From Fig. 23a it is clear that the CO2 plume reaches the +fault zone, crosses it in both aquifers, and distributes partially +along the damage and core zines of the fault. Rightward to +the fault, CO2 flows in the bottom part of the upper aquifer, +and throughout the entire thickness of the lower aquifer, +where the maximum lateral size of the CO2 plume is about +550 m. The shape of the pore pressure plume shown in +Fig. 23b is similar to the previous case, in which bottomhole +pressure equals 350 bar (Fig. 22b): pore pressure diffuses +over the entire reservoir thickness and along the domain of 1 +km length in horizontal direction. Moreover, the volumetric +strain field is qualitatively similar to that obtained in the +previous case (Fig. 22c), while the deformations are larger +by an about an order of magnitude due to the higher pore +pressure (Fig. 23c). Plastic deformations do not develop in +the fault zone, but we observe them in the vicinity of the +perforation interval located in the upper aquifer and in the +bottom part of the salt deposit near the left boundary of the +Preprint submitted to Journal of Natural Gas Science and Engineering +Page 19 of 26 + +-1400 +-1500 +CO2 +salt +-1600 +-1700 +-1800 +anhydrite +-1900 +三 -2000 +upper aquifer, limestone +N-2100 +caprock, dense limestone +-2200 +-2300 +360 m +lower aguifer,limestone +-2400 +basement, +-2500 +-2600 +dense limestone +-2700 +0 +200 +400 +600 +800 +1000 +1200 +1400 +x,myy +七 +yy +xx +xxGeomechanical risks of CO2 storage +Figure 22: Results of coupled hydro-geomechanical modeling of CO2 injection into the upper and lower aquifers at a constant +bottomhole pressure of 350 bar; plots (a) – (c) show the gas saturation, pore pressure increment (compared to the initial +hydrostatic distribution), and volumetric strain fields at the end of the simulation period (30 years), respectively. +Figure 23: Results of coupled hydro-geomechanical modeling of CO2 injection into the upper and lower aquifers at a constant +bottomhole pressure of 600 bar; plots (a) – (c) show the gas saturation, pore pressure increment (compared to the initial +hydrostatic distribution), and volumetric strain fields at the end of the simulation period (30 years), respectively. +formation. The former indicate the possible appearance of +hydraulic fractures near the injector since the bottomhole +pressure exceeds the minimal stress. The main crack does +not open, and despite the increased value of the bottomhole +pressure (600 bar versus 350 bar), we still obtain that the +slip along the fault plane does not occur. The maximum +permeability increase observed in the aquifers is about 2.5%, +while this value in the damage zone of the fault reaches 4%. +Due to the improvement of permeability in the target layers, +the mass of the injected CO2 is higher in the coupled simu- +lation as compared to that obtained using the hydrodynamic +simulation (red line compared to the blue line in Fig. 24). +However, the difference in the injected mass at the end of +the simulation period is insignificant. +4.3.3. Sensitivity analysis of the coupled +hydro-geomechanical model +In the current section, we perform the sensitivity anal- +ysis of the coupled hydro-geomechanical model by vary- +ing mechanical and flow properties of the reservoir. We +analyze the fault stability as well as the development of +plastic deformations leading to the loss of integrity of the +storage domain. The simulations are carried out for realistic +parameters determining CO2 storage in the aquifers of at the +bottomhole pressure of 600 bar. +Figure 24: Dynamics of the CO2 mass injected at a constant +bottomhole pressure 600 bar computed via the coupled hydro- +geomechanical (red line) and hydrodynamic (blue line) models +during the second half of the injection period (15-30 years); +results of simulations using modified coupled models (see +details in Section 4.3.3) are also shown: with reduced values +of the mechanical characteristics in the aquifers and caprock +(green line), with strengthened contrast in the mechanical +properties between the host rock and the fault core in the +fault zone (dashed blue line), with increased permeability in +the fault zone (green line). +We assume that the mechanical properties of both aquifers +and caprock layer between them can vary in certain ranges. +Preprint submitted to Journal of Natural Gas Science and Engineering +Page 20 of 26 + +a) +-1400 +b) +-1400 +c) +-1400 +Ap,bar +43 +-1500 +gas +-1500 +-1500 +-1600 +7.0e-01 +-1600 +1.5e+02 +-1600 +5.0e-04 + 0.6 +-1700 +-1700 +120 +-1700 +0.0004 + 0.5 +-1800 +-1800 +100 +-1800 +0.0003 +0.4 +-1900 +-1900 +80 +-1900 +0.0002 +0.3 +60 +0.0001 + 0.2 +40 +0 +N-2100 +0.1 +N-2100 +20 +N-2100 +-2200 +0.0e+00 +-2200 +0.0e+00 +1.5e-04 +-2300 +-2300 +-2300 +-2400 +-2400 +-2400 +-2500 +-2500 +-2500 +-2600 +-2600 +-2600 +-2700 +-2700 +-2700 +0 +200400 +600 +800 +100012001400 +200 +800 +100012001400 +0 +200 +400 +600 +800 +1000 1200 1400 +x,m +x,m +x,ma) +-1400 +b) +-1400 +Ap,bar +c) +-1400 +-1500 +gas +-1500 +-1500 +-1600 +7.0e-01 +-1600 +4.0e+02 +-1600 +1.5e-03 +0.6 +350 +-1700 +-1700 +300 +-1700 +0.5 +0.001 +-1800 +-1800 +250 +-1800 +0.4 +-1900 +-1900 +200 +0.3 +-1900 +150 +0.0005 +-2000 + 0.2 +100 +N-2100 +0.1 +N-2100 +50 +N-2100 +-0 +-2200 +0.0e+00 +-2200 +0.0e+00 +-2200 +4.0e-04 +-2300 +-2300 +-2300 +-2400 +-2400 +-2400 +-2500 +-2500 +-2500 +-2600 +-2600 +-2600 +-2700 +-2700 +-2700 +0 +200 +400 +600 +800 +100012001400 +0 +200 +400 +600 +800 +100012001400 +0 +200400 +600800 +100012001400 +x,m +x,m +x,mBottomhole pressure 600 bar +1.0e+5 +COUPLEDBASE +COUPLED REDUCED +9.5e+4 +COUPLFD FAUL +COUPLED PERM +9.0e+4 +HYDRO +8.5e+4 +8.0e+4 +ton +7.0e+4 +6.5e+4 +6.0e+4 +5.5e+4 +5.0e+4 +15 +16 +17 +18 +19 +20 +21 +22 +23 +24 +25 +26 +27 +28 +29 +30 +Time, yearGeomechanical risks of CO2 storage +Layer +퐸, GPa +휈 +휌, kg/m3 +푐, MPa +휃 +Upper +aquifer +30 +0.16 +2350 +13 +35 +Caprock +28 +0.25 +2580 +15.3 +26 +Lower +aquifer +22 +0.23 +2360 +11.7 +36 +Table 4 +Decreased values of mechanical properties of the aquifers and +caprock in the realistic formation model shown in Fig. 20. +In Section 4.3.2, we put into FLAC3D simulator their +average values (Table 3). However, one can suggest that +the minimal values of the elastic modulus and strength +parameters can facilitate the undesired mechanical effects +related to the tectonic fault. We carry out the coupled +simulations using the decreased values of the mechanical +parameters outlined in Table 4. +In Fig. 24, we compare the dynamics of the injected +carbon dioxide mass during the second half of the simulation +period obtained by two coupled models: with the average +(red line) and reduced (green line) values of the mechanical +properties (Table 4). We find that in the case of modified +properties the injected mass is larger as compared to than +obtained in the base case. The larger mass of the injected +CO2 is associated predominantly with the larger volumetric +strains (see Fig. 25b) leading to the improvement of porosity +and permeability (see Eq. (12), (13)). In contrast to the base +model, we do not observe the development of plastic defor- +mations near the perforation interval in the upper aquifer, +while they are localized to the lower left part of the salt layer +only. The main fracture opens on asperities along the depth +intervals corresponding to the lower aquifer and the top part +of the upper aquifer. The CO2 plume shape in the modified +model shown in Fig. 25a slightly differs from that obtained +in the base case rightward to the fault. The alteration in the +gas saturation distribution is attributed to the open fracture, +along which a small portion of carbon dioxide flows upward +and then laterally at the top of the upper aquifer. The gas +saturation reaches larger values at the lower aquifer on the +right side of the fault in the modified model. +In Section 4.2, we describe how the mechanical prop- +erties (bulk modulus, shear modulus, cohesion, angle of +internal friction, and dilatancy) are set in the fault zone. All +parameters except for dilatancy decrease by 30% towards the +fault core as compared to the corresponding parameters of +the host rock at the considered depth, while the dilatancy +grows by the same quantity. In the current numerical exper- +iment for the sensitivity analysis, we carry out the coupled +simulations of CO2 injection assuming the alteration of the +mechanical properties in the fault zone by 80% towards its +center. Analyzing the results of simulations we conclude that +there are plastic deformations developed in the fault zone +along its entire length in addition to the similar domains near +the perforation interval in the upper aquifer and in the bottom +left part of the salt layer as in the base case. The main fracture +opens along the entire length; however, its aperture is smaller +as compared to the case of the modified coupled model with +the reduced values of the mechanical properties. We find +that the improvement of permeability in the fault zone due +to the mechanical effects is negligible leading to the same +shapes of the CO2 plume and close values of the injected +mass of carbon dioxide in the modified and base cases. The +comparison of the solutions in terms of the injected CO2 +mass is shown in Fig. 24 (dashed blue line versus red line). +Finally, we conduct a coupled modeling of CO2 injection +into storage aquifers considering the fault zone with the +uniform permeability set to 10 mD so that in the current +experiment, we do not distinguish the damage and core zones +in terms of the flow properties and present the fault zone as +a conduit. The carbon dioxide injected mass obtained using +the modified model exceeds that obtained in the base case +(see 24, orange line compared to the red line). In the modified +model, plastic deformations are developed in the same zones +as in the base case, namely, near the perforation interval in +the upper aquifer and in the bottom part of the salt layer close +to the left border of the formation. Similar to the base case, +the main fracture does not open. The increased permeability +in the fault zone contributes to the flow of a larger volume +of CO2 in parallel and perpendicular directions to the main +crack resulting in the slightly greater size of the carbon +dioxide plume rightward to the fault as compared to the base +case. +5. Summary and conclusions +In this paper, we developed a coupled hydro-geomechanical +model for the simulation of CO2 injection (storage) into +a saline aquifer intersected by a tectonic fault. The model +is based on the reservoir simulator MUFITS, mechanical +simulator FLAC3D, and the in-house algorithm performing +the two-way coupling of the simulators, namely, pressure, +temperature, and density distributions are transferred from +MUFITS to FLAC3D, while porosity and permeability fields +estimated with on the basis of deformations and stresses are +passed from FLAC3D to MUFITS. The latter calculation +relies on the novel mathematical model proposed in the +current study describing the dynamics of the permeability +alteration inside the tectonic fault domain composed of +the damage zone and fault core. During the modeling of +the CO2 injection, MUFITS solves a dynamic problem of +the non-isothermal multiphase flow in a rock formation, +while FLAC3D is applied to solve the quasi-static problem +and computes the mechanical equilibrium of the reservoir. +We simulated the CO2 storage via the developed coupled +approach at the example of two-dimensional synthetic and +realistic reservoir models. +We verified the proposed coupled model by solving the +problem of transient flow of slightly compressible fluid to a +vertical well fully penetrating the infinite-acting reservoir. +Firstly, we compared the results of numerical simulations +with an analytical solution in terms of distributions of pres- +sure, stress tensor components, and vertical displacement +preserving the true diffusivity in the hydrodynamic simula- +tions. Secondly, we accounted for the alteration of porosity +Preprint submitted to Journal of Natural Gas Science and Engineering +Page 21 of 26 + +Geomechanical risks of CO2 storage +Figure 25: +Results of coupled hydro-geomechanical modeling of CO2 injection into the upper and lower aquifers at a fixed +bottomhole pressure of 600 bar and decreased (as compared to base case described in Section 4.3.1) values of the mechanical +properties of the storage aquifers and caprock as descibed in Table 4; plots (a) and (b) show the gas saturation and volumetric +strain fields at the end of the simulation period (30 years), respectively. +and permeability in the hydrodynamic model depending on +the volumetric strain evaluated by the mechanical model. In +this numerical experiment, we compared the results of MU- +FITS+FLAC3D and FLAC3D in terms of the parameters +listed above and a good match is obtained. +Using the synthetic reservoir model, we examined two +locations of the storage aquifer at a depth of 950 m and 2 +km. For each configuration we demonstrated the results of +modeling corresponding to the formation with no tectonic +stresses and at pronounced tectonic stresses applied in the +initial mechanical reservoir state. We observed that in the +absence of the tectonic stresses, plastic deformations do not +develop in the reservoir, and the major fracture in the fault +core opens on asperities in the elastic mode contributing to +an increase in the fault zone permeability along its plane +and CO2 leakage out of the target aquifer. When the tectonic +stresses are pronounced, we found that the plastic deforma- +tions are developed in the fault zone in addition to the major +crack opening. It results in a permeability increase in the +directions along and perpendicular to the fault, CO2 leakage +into the upper aquifer, and considerably larger mass of the +injected carbon dioxide as compared to that obtained in the +case with no tectonic stresses. +In the case of the realistic reservoir, we consider a slice +of the formation sector and analyzed the CO2 injection +with constant bottomhole pressure of 350 bar (base value) +and 600 bar (upper limit). We determined the absence of +undesirable mechanical effects in the base case. For the +increased value of the bottomhole pressure, we demonstrated +that the fault remains stable while plastic deformations are +developed in the vicinity of the perforation interval indi- +cating the possible initiation of hydraulic fractures. Further, +using the realistic reservoir model, we performed the sensi- +tivity analysis of the coupled model to the input parameters +describing the fault behavior. We varied the mechanical +properties of the storage layers and fault zone as well as the +fault permeability. It was shown that the reduced values of +the mechanical properties in the target layers contribute to +an increase in the volumetric deformations (leading to an +increase in porosity and permeability) and a partial opening +of the main fracture. Strengthened contrast in the mechanical +parameters of the host rock and fault core yields insignif- +icant plastic deformations in the fault zone and the main +fracture opening with a negligibly small aperture. Finally, an +increased permeability in the fault zone results in a tangible +increase in the injected CO2 mass. +Acknowledgements +The authors are grateful to the management of Gazprom- +neft Science & Technology Center for organizational and +financial support of this work, in particular to Dr.Sci. Oleg +Ushmaev, Nikolay Glavnov, Evgeny Sergeev and Prof. Mars +M. Khasanov. +A. Constitutive relations to a medium with +internal friction and dilatancy +Prandtl-Rice constitutive relations to a medium with +internal friction and dilatancy are formulated in Nikolaevskii +(1971) as follows: +푑푒푖푗 = Π푖푗푘푙푑휎푘푙, +(46) +where +Π푖푗푘푙 = +[ +− +휈 +2퐺(1 + 휈)훿푖푗훿푘푙 + 1 +4퐺 +(훿푖푘훿푗푙 + 훿푘푗훿푖푙 +)] ++ +1 +4퐻 +( +푁푖푗 + 2 +3Λ훿푖푗 +) ( +푁푘푙 + 2 +3훼훿푘푙 +) +(47) +푁푖푗 = 푠푖푗∕푇 , 푇 = (푠푖푗푠푖푗 +)1∕2 , 푠푖푗 = 휎푖푗 − 훿푖푗휎, 휎 = 1 +3휎푖푖 +Here, 푑푒푖푗 and 푑휎푘푙 are components of strain and stress +tensor increments; 퐺 and 휈 are shear modulus and Poisson +coefficient, respectively; 푠푖푗 are components of deviatoric +Preprint submitted to Journal of Natural Gas Science and Engineering +Page 22 of 26 + +a +-1400 +b) +-1400 +-1500 +-1500 +-1600 +7.0e-01 +-1600 +1.5e-03 +0.6 +-1700 +-1700 +0.5 +0.001 +-1800 +-1800 +0.4 +-1900 +0.3 +-1900 +0.0005 +三 -2000 += -2000 +0.2 +N-2100 +0.1 +N-2100 +0 +-2200 +0.0e+00 +-2200 +4.0e-04 +-2300 +-2300 +-2400 +-2400 +-2500 +-2500 +-2600 +-2600 +-2700 +-2700 +0 +200 +400 +600 +800 +1000 +12001400 +0 +200 +400 +600 +800 +1000 +1200 +1400 +x,m +x,mGeomechanical risks of CO2 storage +stress tensor and 푇 is the shear stress intensity; Λ is dilatancy +coefficient; 훼 is internal friction coefficient; in the tensor ex- +pressions formulated above we use the standard convention +on summation over repeating indexes. +The alternative form of Eqs. (46) and (47) is formulated +in Rudnicki and Rice (1975): +Δ휎푖푗 = 퐸푖푗푘푙Δ휀푘푙, +(48) +퐸푖푗푘푙 = 퐺 +{[(훿푖푘훿푗푙 + 훿푖푙훿푘푗 +)+ +(퐾 +퐺 − 2 +3 +) +훿푘푙훿푖푗 +] +− +− +퐺 +(퐻 + 퐺) + 훼Λ퐾 +( +푁푖푗 + 퐾 +퐺 Λ훿푖푗 +) +(푁푘푙+ 퐾 +퐺 훼훿푘푙) +} +, (49) +where 퐾 = 2(1 + 휈)퐺∕[3(1 − 2휈)]is the bulk modulus. +B. Implementation of the fault zone +permeability into the hydrodynamical +model +In the main text, we introduce three permeability types: +1. permeability of the host rock – 푘, Eq. (13), +2. permeability of the system of the natural fractures in +the damage zone of the fault – 푘푓, Eq. (28), +3. permeability of the main fracture opened on asperities +– 푘푐, Eq. (41). +In the current section, we describe how parameters 푘, 푘푓, +and 푘푐 are combined in the hydrodynamical model. +We begin with the cells related to the damage zone +(Fig. 26a). Each cell includes two interpenetrating isotropic +Figure 26: +The figure shows the representations of the cells +belonging to the damage zone of the tectonic fault (panel a) +and to its core (panel b). +continua, namely, host rock and natural fractures. We apply +the equation describing the total permeability of the layered +formation in the direction parallel to the stratification: +̄푘 = +∑ +푖 푘푖ℎ푖 +∑ +푖 ℎ푖 +, +(50) +where 푘푖 and ℎ푖 are the permeability and the thickness of +each layer in the layered reservoir. As a result, we estimate +the permeability of the cells located in the damage zone of +the fault as follows: +푘푥 = 푘푧 = 푘(1 − 휀푝) + 푘푓휀푝, +(51) +where 휀푝 is plastic volumetric strain. +Next, we move to the cells related to the fault core +(Fig. 26b). Each of them is intersected by a main crack. In +the derivations, we do not account for the fracture inclination +assuming that the tectonic fault is approximately vertical and +parallel to z-axis. 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Proceedings of the National +Academy of Sciences 109, 10164–10168. +Preprint submitted to Journal of Natural Gas Science and Engineering +Page 26 of 26 + diff --git a/RNE4T4oBgHgl3EQfKgzs/content/tmp_files/load_file.txt b/RNE4T4oBgHgl3EQfKgzs/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e11618cbd50dbac918b6922dd51a0d14c05a00b5 --- /dev/null +++ b/RNE4T4oBgHgl3EQfKgzs/content/tmp_files/load_file.txt @@ -0,0 +1,2173 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf,len=2172 +page_content='CO2 storage in deep saline aquifers: evaluation of geomechanical risks using integrated modeling workflow Evgenii Kanina,∗, Igor Garagasha,b, Sergei Boronina, Svetlana Zhigulskiye, Artem Penigine, Andrey Afanasyevd, Dmitry Garagashc and Andrei Osiptsova aProject Center for Energy Transition and ESG, Skolkovo Institute of Science and Technology (Skoltech), Bolshoy Boulevard 30, bld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 1, Moscow 121205, Russian Federation bInstitute of Physics of the Earth, Russian Academy of Scienses, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Gruzinskaya st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 10, Moscow 123995, Russian Federation cDepartment of Civil and Resource Engineering, Dalhousie University, 1360 Barrington Street, Halifax B3H 4R2, Nova Scotia, Canada dInstitute of Mechanics, Moscow State University, Michurinsky Pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 1, Moscow 119192, Russian Federation eGazpromneft Science & Technology Center, 75-79 liter D Moika River emb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', St Petersburg 190000, Russian Federation A R T I C L E I N F O Keywords: CO2 storage reservoir simulator mechanical simulator tectonic fault fault slip plastic deformations integrity loss CO2 leakage A B S T R A C T CO2 injection into a saline aquifer crossed by a tectonic fault is studied with coupled fluid mechanics geomechanics modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The model is based on reservoir simulator MUFITS and mechanical simulator FLAC3D linked by the in-house algorithm of the data exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' MUFITS simulates the non-isothermal multiphase flow of CO2 and brine in rock formation accounting for phase transitions and thermal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The modeling workflow is sequential, so that hydrodynamical simulations are carried out at a certain time interval, after which pressure, temperature, and density distributions are passed to FLAC3D, which calculates the equilibrium mechanical state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Computed deformations and stresses are utilized to update the porosity and permeability fields for the subsequent hydrodynamic modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In particular, we focus on the tectonic fault and its behavior during CO2 injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We distinguish the damage zone and core inside the fault and derive the closure relations for the their permeability alteration analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The developed coupled approach is applied to simulate CO2 injection into synthetic and realistic reservoir models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' For the former one, we study the effect of formation depth and presence of the tectonic stresses at the initial mechanical state, while for the latter, we consider different injection modes (bottomhole pressure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In each numerical experiment, we describe the evolution of the fault permeability due to the slip along its plane and the development of plastic deformations leading to the loss of reservoir integrity and CO2 leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Sensitivity analysis of the coupled model to realistic values of input parameters to assess the fault stability is carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Introduction Carbon dioxide (CO2) is a greenhouse gas contained in the atmosphere, which forms predominantly due to burning of fossil fuels as a result of human activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Continuous growth in CO2 concentration in the atmosphere intensify the greenhouse effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Consequently, the mean temperature grows contributing to an increase in the frequency and severity of catastrophic climate events (Pörtner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Emission of carbon dioxide can be reduced by using low- carbon fuels and improving energy efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' However, application of these techniques is insufficient to reduce the concentration of CO2 significantly and prevent mean temper- ature growth (IEA, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Resolving this issue requires the development and application of CO2 sequestration technolo- gies (Kazemifar, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' They include carbon dioxide capture at emission sources, its liquefaction, and transportation to fields where CO2 is injected into deep saline aquifers or depleted oil/gas formations in the supercritical state (Bickle, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' During the injection stage, CO2 displaces pore fluid (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', natural gas or water) and forms a plume propagating due to the pressure difference between the injector and far- field pore pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In addition, the buoyancy force affects the plume dynamics since carbon dioxide is usually lighter ∗Corresponding author evgenii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='kanin@skoltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='ru (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Kanin) as compared to pore fluids (Bryant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' therefore, a target reservoir for secure CO2 storage have to be covered by a low-permeable caprock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Flow of CO2 in rock formation is accompanied by various phase transitions and chemical reactions including dissolution of carbon dioxide in brine as well as water vapor in CO2 (Huppert and Neufeld, 2014), precipitation of insoluble salts due to interactions of carbon- ated water with minerals composing the rock (de Coninck and Benson, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' De Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Development of an efficient technology of CO2 storage in underground formations includes the solution of several problems, namely, (i) evaluation of the formation capacity or the maximum CO2 volume that can be injected into the geological formation (Bachu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Bradshaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' De Silva and Ranjith, 2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (ii) estimation of the reservoir injectivity, which is the maximum injection rate of carbon dioxide based on operating param- eters, rock permeability, properties of CO2 and pore fluid (Rutqvist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Stauffer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Burton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Mathias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (iii) assessment of geomechan- ical effects leading to undesired events (Lucier and Zoback, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Newmark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Zoback and Gorelick, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Rutqvist, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' White and Foxall, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Gholami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The current work is devoted to the last problem, namely, evaluation of geomechanical risks of CO2 injection into underground formations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Preprint submitted to Journal of Natural Gas Science and Engineering Page 1 of 26 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='04931v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='geo-ph] 12 Jan 2023 aGeomechanical risks of CO2 storage Three types of undesirable phenomena related to ge- omechanical effects are identified (Hawkes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Pawar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2016): (i) loss of CO2 storage integrity and leakage of carbon dioxide out of the target aquifer to upper layers, (ii) activation of tectonic faults and fractures intersecting the storage reservoir near the overpres- sured zones, (iii) development and uncontrolled growth of hydraulic fractures due to overpressured and cooled zone around the injector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (iv) deformation of the Earth surface located above the storage area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' These events can contribute to the seismic activity (earthquakes) in the neighbor regions, pollution of fresh water aquifers due to leakage of CO2 or pore brine out of the target layer as well as damage to surface infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' When the reservoir is intersected by a tectonic fault, and the CO2 plume at high pore pressure reaches it, the effective normal stress at the fault plane declines leading to fault activation (Streit and Hillis, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Konstantinovskaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The slip promotes not only seismic activity and earthquakes but also the enhancement of the fault zone permeability (Rinaldi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2014, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Guglielmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2017, 2021), which can lead to opening of the major crack in the fault core and natural fractures in the damage zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Consequently, the tectonic fault forms a conduit along which CO2 can flow out of the target reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Summarizing the outlined geomechanical risks, we would like to highlight that it is crucial to model accurately CO2 storage in underground reservoir during the planning stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Mathematical modeling can be carried out to estimate the favorable injection regimes to prevent undesirable geomechanical phenomena running in the target reservoir and surrounding layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The problem of CO2 injection into deep geological for- mation includes coupled thermal-hydraulic-mechanical pro- cesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Two types of mathematical algorithms are devel- oped to solve the set of governing equations: fully coupled and sequentially coupled (Dean et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Kim, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Ferronato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Rutqvist, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The former approach involves the simultaneous solution of the thermal-hydraulic and mechanical sets of governing equations, which is computationally intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In the lat- ter method, the governing equations of the sub-problems are solved sequentially, and the required data is passed in between solvers via the coupling algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The sequen- tially coupled approach is less computationally expensive as compared to fully coupled one and it is more flexible in terms of calculation algorithm as it allows one to (i) choose the degree of coupling in between thermal-hydraulic and mechanical models, namely, every iteration at current time step (iterative coupling), a single coupling procedure at each time step (non-iterative coupling);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (ii) the coupling frequency, which are time intervals, at which the coupling is applied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (iii) to utilize different spatial domains and time- stepping algorithms in each sub-problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (iv) to utilize the existing open-source codes and commercial simulators as hydrodynamical and/or mechanical solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' If the numerical solution converges, both approaches provide identical results of simulations as discussed by Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Coupled TOUGH-FLAC simulations is an example of the sequential approach applied to the solution of the hydro-geomechanical problem described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' It is based on the hydrodynamics simulator TOUGH (Transport of Unsaturated Groundwater and Heat, (Pruess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 1999)) solving non-isothermal, multicomponent, multiphase flow sub-problem and mechanical simulator FLAC3D (Fast La- grangian Analysis of Continua in 3D, (Itasca, 1997)) dealing with the geomechanical sub-problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Coupled TOUGH- FLAC simulations are proposed in (Rutqvist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Rutqvist and Tsang, 2003), and, since then, its capabilities have been extended considerably (Rutqvist, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Blanco- Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Rinaldi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The simulator was applied to investigate various problems related to CO2 storage including the tightness of the caprock, maximum sustainable injection pressure, thermal effects, tectonic fault reactivation, induced seismicity as well as the risk of CO2 and pore fluid leakage along the fault zone, (Rutqvist and Tsang, 2002, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Rutqvist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2008, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Cappa and Rutqvist, 2011, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Rinaldi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2014, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Vilarrasa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Guglielmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Luu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Besides FLAC3D, the TOUGH family codes were cou- pled with other geomechanical simulators and packages, in particular, the open-source library PyLith (Aagaard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2013) as demonstrated by Blanco-Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Var- ious TOUGH-based geomechanical models are also sum- marized in a review paper (Rutqvist, 2017) and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Rutqvist (2011) provided an overview of reservoir and mechanical simulators applied for modeling of coupled thermal–hydraulic–mechanical processes in geological for- mations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We would like to mention that certain software packages, namely, CMG (module GEM, CMG (2018)) and CODE-BRIGHT (Olivella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 1994, 1996) allow simulat- ing CO2 storage accounting for a non-isothermal multiphase flow of carbon dioxide and pore fluids accompanied by ge- omechanical effects within the frame of the single simulator (Vilarrasa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Jahandideh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In the current study, we develop a coupled hydro- geomechanical model of CO2 storage in a saline aquifer intersected by a single tectonic fault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The model is based on the academic reservoir simulator MUFITS (Afanasyev, 2020), commercial mechanical simulator FLAC3D (Itasca, 1997), and our in-house coupling algorithm of data exchange in between simulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Furthermore, we propose an analyti- cal model describing permeability evolution in the tectonic fault zone of rock formation and embed it into the coupled modeling workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The closure relations include several parameters, which can be evaluated in lab experiments and through the solution of the inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The latter one is tuning of input parameters to approximate the field observations by the results of simulations using the coupled hydro-geomechanical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Our main goal is to study the fault behavior during CO2 injection at different bottomhole pressure values, formation configurations, stress conditions, Preprint submitted to Journal of Natural Gas Science and Engineering Page 2 of 26 Geomechanical risks of CO2 storage and reservoir properties to evaluate the associated geome- chanical risks including fault activation and CO2 leakage out of the target aquifer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We organize the paper in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Section 2 describes the coupled hydro-geomechanical model based on the reservoir simulator MUFITS, mechanical simulator FLAC3D, and the algorithm executing the data transfer between simulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In Section 3, we derive the analytical relations governing the permeability evolution in the damage zone and the core of the tectonic fault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Section 4 presents the results of CO2 storage simulations based on synthetic and realistic reservoir models supplemented by the discussion with the focus on geomechanical risk assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Finally, we summarize the findings of the paper in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Modeling approach We develop a coupled hydro-geomechanical model to evaluate the geomechanical risks associated with CO2 se- questration in a deep saline aquifer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The model is based on freely distributed reservoir simulator MUFITS (Afanasyev (2020), MUFITS – Reservoir Simulation Software) and commercial mechanical simulator FLAC3D (Itasca, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Hydrodynamical model We employ the reservoir simulator MUFITS (Afanasyev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Afanasyev and Vedeneeva, 2021) for modeling the Darcy flow in rock formation caused by the injection of CO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We set the software to account for the dissolution of CO2 in brine and the presence of water vapor in the gas phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The brine salinity is reduced to the NaCl concentra- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Furthermore, we account for the temperature changes caused by the Joule-Thomson effect in the supercritical CO2, the convective heat transfer and heat conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Thus in our study,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' the non-isothermal flow of CO2–H2O–NaCl fluid is governed by the following equations 휕 휕푡 ( 휙 ∑ 푗=푔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='푙 휌푖푐푖(푗)푠푖 ) + ∇ ⋅ ( ∑ 푗=푔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='푙 휌푖푐푖(푗)퐮푖 ) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 푗 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 3 (1) 퐮푖 = −퐤푘푟푖 휇푖 (∇푃푖 − 휌푖퐠) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 푖 = 푔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 푙 (2) 휕 휕푡 ( 휙 ∑ 푖=푔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='푙 휌푖푒푖푠푖 + (1 − 휙)휌푟푒푟 ) + ∇ ⋅ ( ∑ 푖=푔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='푙 휌푖ℎ푖퐮푖 − 휆∇푇 ) = 0 (3) Ω (푃푔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 푇 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 푐푙(3) ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Ω = {휌푖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 푒푖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' ℎ푖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 휇푖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 푐푖(푗)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 푒푟 = 퐶푟푇 (4) 푘푟푖 = 푘푟푖(푠),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 푃푔 − 푃푙 = 푃푐푔푙(푠) (5) 푠푔 + 푠푙 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 3 ∑ 푗=1 푐푖(푗) = 1 (6) where 휙 is the porosity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 휌 is the density,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 푐푖(푗) is the 푗th com- ponent mass fraction in brine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 푠 is the fluid saturation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 푢 is the Darcy velocity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' k = diag(푘푥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 푘푦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 푘푧) is the permeability tensor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 푘푟푖 is the relative permeability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 휇 is the fluid viscosity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' g is the gravity acceleration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 푒 and ℎ are the specific energy and enthalpy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 휆 is the heat conductivity of saturated porous medium,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 푇 is the temperature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 푃푐푔푙 is the capillary pressure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' and 퐶푟 is the specific heat capacity of rock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The subscripts 푔, 푙 and 푟 refer to the parameters of the gas, liquid (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' brine) and rock phases, while subscript (푗) denotes the parameters of the 푗th component, where 푗 = 1, 2 and 3 correspond to CO2, H2O and NaCl, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Equations (1) and (3) are the mass and energy balance equations and (2) is Darcy law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' These balance equations are supplemented by the the saturation functions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (5) and the closing relations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The changes in temperature governed by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (3) and (4) can be significant in the regions of rapid changes in pressure and due to the difference between the reservoir temperature and that of the injected gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Besides the fluid properties, the thermal effects can also influence the stresses and deformations induced by the injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Therefore, we track these effects by simulating non-isothermal flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Equation (4) shows schematically the equations used for modeling the fluid phase equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We assume that NaCl is present only in brine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Thus, its concentration in gas is zero (푐푔(3) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (4), the parameters of the fluid including the phase densities and viscosities are parametrized as functions of the pressure, temperature, and the brine salinity 푐푙(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Generally, we follow the methodology of Spycher and Pruess (2005) and Pruess and Spycher (2007) for predicting the fluid properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Here, we avoid further complications of the hydrodynamic model that can also ac- count for halite precipitation near the injection well (Pruess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In our simulations, we do not account for pore water salinity resulting in the absence of the effects related to salt precipitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Therefore we present a simplified version of the governing equations to keep the presentation short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The relative permeability and capillary pressure curves in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (5) are given by 푘푟푔 = (1 − ̂푆)2(1 − ̂푆2), ̂푆 = 푠푙 − 푠푙푟 1 − 푠푙푟 − 푠푔푟 푘푟푙 = √ 푆∗ [ 1 − (1 − (푆∗)1∕휆푙)휆푙]2 , 푆∗ = 푠푙 − 푠푙푟 1 − 푠푙푟 푃푐푔푙 = −푃푐0 ( 푠−1∕휆푐 푙 − 1 )1−휆푐 (7) where 푠푙 is the saturation of liquid phase, 푠푔 is the saturation of gas phase, 푠푙푟 is the irreducible saturation of brine, 푠푔푟 is the irreducible saturation of gas, 휆푙, 휆푐 are the exponents, 푃푐0 is the strength coefficient, correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Gas relative permeability is taken from (Corey, 1954), while liquid rel- ative permeability and capillary pressure are proposed by Van Genuchten (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Preprint submitted to Journal of Natural Gas Science and Engineering Page 3 of 26 Geomechanical risks of CO2 storage 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Mechanical model We utilize simulator FLAC3D for computing the me- chanical equilibrium state of the fluid-saturated formation in terms of stresses and deformations corresponding to pre- defined pore pressure, temperature, and density distribu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' While the hydrodynamics simulator MUFITS solves the dynamic problem, the mechanical simulator FLAC3D deals with a static one in the framework of the quasi-static approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Two constitutive models implemented in FLAC3D are utilized in the present study, namely, linear elastic isotropic model based on Hooke’s law and elastoplas- tic one based on the Drucker-Prager yield condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' FLAC3D calculates the distributions of stresses and deformations via the numerical solution of the system of governing equations formulated with respect to mechanical (stresses) and kinematic (strain increment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' velocity) vari- ables as follows: 휌푑퐯 푑푡 = ∇ ⋅ 흈 + 휌퐠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (8) 휺 = 1 2 (∇퐮 + (∇퐮)푇 ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' ̇휺 = 1 2 (∇퐯 + (∇퐯)푇 ) (9) Δ흈′ = \ue232(흈′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' ̇휺Δ푡),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (10) where 휌 is the saturated rock density,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 퐯 is the velocity distribution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 흈 is the stress tensor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 휺 is the infinitesimal strain tensor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 퐮 is the displacement distribution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' ̇휺 is the infinitesimal strain rate tensor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 흈′ = 흈 + 훼푃 is the effective stress tensor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 푃 is the pore pressure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 훼 is the Biot coefficient,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' \ue232 denotes material functions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Δ푡 is the time increment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The strain increment, Δ휺, can be represented by the sum of elastic, plastic, and thermal parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Equation (8) describes momentum conservation law, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (9) are Cauchy relations, while Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (10) are constitutive relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' FLAC3D approximates the deformable solid medium by elementary tetrahedrals, and their behavior is governed by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (8)-(10) in accordance with the applied forces and boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The system of equations is solved for the specified geometry and material properties at prescribed boundary and initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Below we describe several features of numerical solution to governing equations as implemented into FLAC3D: The finite difference technique is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The first order time and spatial derivatives are represented by the finite differences based on the assumption that the variables alter linearly over a spatial segment and throughout a time interval;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The continuous medium is replaced by an equivalent discrete one, in which all forces (external and internal) are evaluated at the nodes of the three-dimensional grid used to approximate the deformable medium;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The dynamic solution approach is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The inertial terms in the equation of motion are utilized as an indi- cator for the asymptotic approximation of the system mechanical equilibrium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Within the described framework, the motion equation for the continuous medium is transformed into the discrete form of Newton law formulated at the grid nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The sys- tem of ordinary differential equations is solved numerically using an explicit finite-difference time advance scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The definition of the strain rate tensor through the velocities at nodes includes the spatial derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' For each time step, the calculation procedure is as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' calculation of the updated deformations based on the velocities approximated at grid nodes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' computation of the stresses using the deformations, stresses at the previous time moment, and constitutive relations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' update the velocities and displacements based on the motion equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The sequence 1-3 is repeated at each internal FLAC step, at which the maximum unbalanced force is evaluated at the grid nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' When the force becomes less as compared to the tolerance value, the mechanical system is assumed to be in the equilibrium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' When the unbalanced force reaches a constant non-zero value, it means that the entire system or its part is in the steady-state plastic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Calculations can be interrupted at any FLAC step to analyze the solution behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' For the convenience, we utilize the thermoelastic model, in which stresses and deformations are linked with the tem- perature 푇 distribution only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Here, the temperature equals to the sum of the actual temperature 푇 actual and pseudo- temperature 푇 pseudo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The latter one is introduced to describe the effect of the pore pressure 푃, and it can be determined from the analogy of poroelastic and thermoelastic problems: 푇 pseudo = 훼푃 3휅퐾 , (11) where 휅 is the thermal-expansion coefficient, and 퐾 is the bulk modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Hence, stresses and deformations corre- sponding to the pressure 푃 and temperature 푇 actual fields can be calculated by the thermoelastic model where the formation is heated up to the temperature 푇 = 푇 actual + 푇 pseudo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In the current work, the Biot coefficient is set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Coupling algorithm The governing equations implemented in the hydrody- namical simulator MUFITS and the mechanical simulator FLAC3D are solved sequentially, and the coupling param- eters are transferred between simulators at certain time in- stants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In the current study, we consider identical spatial meshes for hydrodynamical and mechanical simulations and implement the “in-house” algorithm for the data exchange between the simulators using on the approach proposed by Rutqvist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Coupling procedure is summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' During time interval 푡 ∈ [푡푖−1, 푡푖], MUFITS carries out the simu- lation of the multiphase Darcy flow with account of thermal effects at the current rock porosity 휙(퐫, 푡푖−1) and perme- ability 푘(퐫, 푡푖−1) distributions providing the pore pressure Preprint submitted to Journal of Natural Gas Science and Engineering Page 4 of 26 Geomechanical risks of CO2 storage 푃(퐫, 푡푖), temperature 푇 (퐫, 푡푖), and saturated rock density 휌(퐫, 푡푖) fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The later parameter is calculated as follows: 휌 = [휌푙푆 + 휌푔(1 − 푆)] 휙 + 휌푠(1 − 휙), where 휌푙 is the liquid phase density (water with/without dis- solved CO2), 휌푔 is the gas phase density (CO2 with/without dissolved water vapor), 푆 is the liquid phase saturation, and 휌푠 is the rock density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Figure 1: Schematic representation of MUFITS and FLAC3D coupling performing hydro-geomechanical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Then, the hydrodynamical simulation is paused, and the calculated fields 푝(퐫, 푡푖), 푇 (퐫, 푡푖), 휌(퐫, 푡푖) are passed to the mechanical simulator FLAC3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We should note that pore pressure and temperature are approximated in the centers of the mesh cells in MUFITS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' However, in FLAC3D, tem- perature (11) is approximated in the mesh nodes so that an interpolation procedure is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' FLAC3D computes the stress state 흈(퐫, 푡푖) and deformations 휺(퐫, 푡푖), where 흈, 휺 are stress and infinitesimal strain tensors, corresponding to the mechanical equilibrium at the pore pressure 푃(퐫, 푡푖), temperature 푇 (퐫, 푡푖), and density 휌(퐫, 푡푖) fields based on the embedded geological model of the formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Coupling algorithm updates the reservoir porosity field 휙(퐫, 푡푖)using the total volumetric strain distribution 휀푣(퐫, 푡푖) = tr 휺(퐫, 푡푖) (both plastic and elastic), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', the trace of the strain tensor, calculated by FLAC3D and according to the relation (Chin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2000): 휙 = 1 − (1 − 휙0)푒−휀푣, (12) where 휙 and 휙0 are the current and initial porosity values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The permeability field 푘(퐫, 푡푖) is found using the power-law relation (van Golf-Racht, 1982): 푘 = 푘0 ( 휙 휙0 )푛 , (13) where 푘 and 푘0 denote the current and initial permeability values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The power-law exponent varies typically between 3 and 8 (Yehya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2018), and we take 푛 = 5 for the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In the current study, we pay particular attention to the fault zone and permeability alteration here due to the slip along the fault plane and plastic deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We discuss the aspects linked with the fault zone in Section 3 and describe the embedding of its permeability into the hydrodynamical model in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' As a result, we estimate the changes in the rock porosity and permeability corresponding to the pore pressure, tem- perature variations, and the tectonic fault state contributing to both elastic and plastic deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Next, updated dis- tributions 휙(퐫, 푡푖) and 푘(퐫, 푡푖) are passed back to MUFITS simulator (no interpolation is required since 휀푣 values are defined in the centers of the mesh cells similar to porosity and permeability), where the hydrodynamic simulation con- tinues for the next time interval 푡 ∈ [푡푖, 푡푖+1] on which 휙 and 푘 are fixed and equal 휙(퐫, 푡푖), 푘(퐫, 푡푖), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Alteration of permeability of the rock with a tectonic fault due to variations in pore pressure Injection of CO2 into an underground formation is ac- companied by a local change in pore pressure and temper- ature leading to the alteration of the stress state, which can contribute to the activation of a tectonic fault located at a certain distance from the well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' One can note that pressure perturbation propagates faster than the CO2 plume meaning that the activation of faults and fractures can occur not only in the vicinity of the injection well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Moreover, fault slip and rock deformations in the fault zone improve its permeability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Therefore, a coupled hydro-geomechanical model of CO2 injection has to describe the permeability modification due to changes in pore pressure and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In this section, we present mathematical sub-models implemented into the coupled model (Section 2), which allow us to describe the corresponding geomechanical effects and risks: (i) activation of the tectonic fault and slip along its plane due to variation in a stress state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (ii) alteration of the fault zone permeability due to opening of the major crack on asperities in the core and natural fractures in the damage zone;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (iii) disintegration of CO2 storage zone and carbon dioxide leakage to the upper collectors along the fault zone being a highly permeable conduit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Effect of inelastic deformations along the principal fault slip line on permeability of damage rock zone Fault zones determine many processes running in the Earth crust and affect its mechanical properties significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Tectonic faults are most active structural formations through which an energy exchange in between tectonic blocks is carried out (Rice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Kanamori and Rivera, 2006), they also play a significant role in underground fluid move- ment (Townend and Zoback, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Fault-block structure of Preprint submitted to Journal of Natural Gas Science and Engineering Page 5 of 26 Pressure, temperature, density Hydrodynamical Mechanical model model Multiphase flow, Stresses, phase transitions, displacements, temperature effects deformations Porosity and permeabilityGeomechanical risks of CO2 storage Earth crust and sedimentation layer is one of key factors determining the stress state of underground formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Fault structure Fault structure usually includes relatively narrow zone of large deformations surrounded by the transitional zone of fractured rock usually named as damage zone (Wibberley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The damage zone is surrounded by a host rock (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Figure 2: Conceptual representation of the fault structure, after Shipton and Cowie (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' damage zone and fault core are shown by DZ and FC, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The damage zone width is determined by a set of param- eters including the thickness of rock layer being deformed, fault length and the density of fractures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Quantitative evalu- ation of damage zone is usually carried out by determining the density of rock fractures as a function of distance to the fault slip line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' According to existing studies, for rocks with low porosity, there is an exponential decrease in density of fractures with an increase in the distance to the fault slip line (Anders and Wiltschko, 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Vermilye and Scholz, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Mitchell and Faulkner, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' For fault zones in high-porosity rocks, the distribution of fracture density in the vicinity of the fault core is not clear: exponential law is confirmed in some cases (Anders and Wiltschko, 1994), while in other cases no correlation can be established (Shipton and Cowie, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Correlations in between the width of damage zone and key fault parameters (fault slip, length, fault displacement, throw) were discussed by geologists for several decades (Wibberley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Faulkner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Existing correlations differ significantly, and the reason is that authors define the width of damage zone differently, for example, it can be (i) a length measured from one the sides of the fault core;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (ii) a single measurement or mean of the mea- surements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (iii) maximum width of the zone confined by the damage zone envelope (Shipton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The structure of rock damage zone and fault core are closely related with the permeability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Caine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (1996) identified four types of the fault zone permeability structure based on the analysis of studies (Chester and Logan, 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Forster and Evans, 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Moore and Vrolijk, 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Newman and Mitra, 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Depending on dimensions and structure elements of the fault core and damage zones it was suggested to consider faults with localized conduits, distributed con- duits, combined conduit-barriers and localized barriers as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 3 (Caine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Localization of inelastic deformations Tectonic faults are zones of localized irreversible de- formations so that their initiation can be considered as a result of bifurcation of deformation process developed due to rheological instability of Earth crust (Rudnicki and Rice, 1975;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Garagash and Nikolaevskii, 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Development of instability is associated with fracturing and dilatancy of rock under applied stress as well as with the effect of pressure on inelastic deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Laboratory experiments on rock samples showed that their deformation is accompanied by the development of existing microfractures and pores as well as initiation of new fractures leading to alteration of effective mechanical properties of rock (Nikolaevskii et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Paterson and Wong, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' This process depends both on applied stress and interaction of fracture walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Its distinctive feature is dilatancy, which is irreversible increase in rock volume due to increase in size of pores and fracture aperture (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The most intensive change in rock structure occurs in the vicinity of peak stress before initiation of microscopic fracture-like defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Internal instability in the framework of non-associated plastic deformation law Inelastic deformation of rock is carried out due to slip along existing fractures and initiation of new defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The process can be described using exiting laws of plastic de- formation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', in the Prandtl-Rice form, accounting for the dilatancy and interaction of fracture walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Generalization to Prandtl-Rice constitutive relations to a medium with internal friction and dilatancy is first for- mulated in (Nikolaevskii, 1971) in the form of dependence of deformation increment on stress increment, while its alternative form (stress increment in terms of deformation increment) is given by Rudnicki and Rice (1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We pro- vide these relations in Appendix A for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' These relations are used by Rudnicki and Rice (1975) to study localization of inelastic deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' It was found that under the Mohr-Coulomb limiting condition 푇 = 푐 − 훼휎 (14) the formation of ordered structure of fractures occurs at the critical value of plastic hardening modulus 퐻푐푟 expressed as 퐻푐푟 퐺 = 1 + 휈 9(1 − 휈) (훼 − Λ)2 − 1 + 휈 2 (휏2 푇 + 훼 + Λ 3 )2 (15) In equations (14), (15), 푇 is the shear stress intensity, 푐 is the rock cohesion, 휎 is the mean stress, 퐺 and 휈 are shear modulus and Poisson’s ratio, respectively, Λ is dilatancy coefficient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 훼 is internal friction coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Preprint submitted to Journal of Natural Gas Science and Engineering Page 6 of 26 finite DZ width constant max at fault tip deformation deformation density in DZ density increases along FC DZ width increases with displacement maximum FC width is constant host host damage zone envelope DZ slip-surfaces FC DZ within DZGeomechanical risks of CO2 storage Figure 3: Conceptual scheme of tectonic fault permeability, after Caine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Figure 4: Diagram measured during triaxial stress loading of rock sample in laboratory conditions, after Jaeger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Formed fractures are ordered along the axis of mean principal stress 휎2 at the angle 휓 with respect to direction of principal confining stress 휎3: 휓 = arctan √ 3[(1 − 휈)휏2 + 휏3] − (1 + 휈)(훼 + Λ)푇 (1 + 휈)(훼 + Λ)푇 − 3[(1 − 휈)휏2 + 휏1] (16) where 휏1 > 휏2 > 휏3 are principal components of the deviatoric stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Evaluation of horizontal tectonic stress and dilatancy cofficient We evaluate the dilatancy coefficient Λ and horizontal stress leading to formation of a fault inclined by angle 휓 under certain parameters of rock strength, namely, cohesion 푐 and internal friction coefficient 훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Mohr-Coulomb limiting condition is formulated for fluid-saturated porous medium as follows: 푇 + 훼휎 = 푐 − 훼푝푓, (17) where 푝푓 is the pore pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We consider the elastic rock layer located at the depth ℎ under vertical stress 휎33 and horizontal stresses 휎11 = 휎푒 11 + 휎푡 11, 휎22 = 휎푒 22 + 휎푡 22, (18) due to lateral rock repulsion 휎푒 11 and 휎푒 22 as well as tectonic stresses 휎푡 11, 휎푡 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Total horizontal stresses due to lateral repulsion in elastic fluid-saturated layer are expressed as follows (Eaton (1969)): 휎푒 11 = 휎푒 22 = 휈 1 − 휈 휎33 − 푝푓 1 − 2휈 1 − 휈 (19) Expression (19) was obtained in the absence of thermal- induced stresses in the rock formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Temperature of the rock increases with an increase in the depth according to geothermal gradient, which depends on rock composition, thermal conductivity and density of heat flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Usually geothermal gradient takes values in the range between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='5 C up to 20 ◦C with the average value of 3 ◦C for 100m depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Constitutive relation for a heated elastic layer has the following form (Timoshenko and Goodier, 1970) 휎푖푗 = 2퐺휀푖푗 + 2퐺휈 1 − 2휈 휀훿푖푗 − 2휅퐺(1 + 휈) 1 − 2휈 푇푓훿푖푗 (20) where 푇푓 is the temperature and 휅 is the thermal expansion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Following the derivation of expressions (19) we consider elastic half-space with temperature distribution along the depth 푇푓 = 푇푓(푥3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In this case, the deformations are expressed as follows 휀11 = 휀22 = 0, 휀33 = 휅푇푓 1 + 휈 1 − 휈 (21) so that equilibrium equations 휎푖푗,푗 = 0 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Now stress components according to (20) are expressed as follows: 휎11 = 휎22 = −2휅푇푓퐺1 + 휈 1 − 휈 , 휎33 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (22) Expressions (22) allow to generalize Eaton expressions (19) for horizontal stresses in a heated fluid-saturated half- space as follows: 휎푒 11 = 휎푒 22 = 휈 1 − 휈 휎33 − (푝푓 + 푝푡)1 − 2휈 1 − 휈 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='(23) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='Preprint submitted to Journal of Natural Gas Science and Engineering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='Page 7 of 26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='Distributed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='Caombined ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='Conduit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='low ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='ECOTC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='high ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='Conduif-Barrier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='higl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='Acerctionary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='high ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='Dixic Valley ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='Prisins ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='Fault ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='Damage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='Permeability Structures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='Lunage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='In Fault Zones ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='2u07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='Shawuogunk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='San Gabnel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='low ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='Mountains ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='Cataclasites ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='low ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='Localized ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='Locatized ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='Conduit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='low ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='+ Core ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='high ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='Barrier300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='Quartzite ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='Axial stress (MPa) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='Axial strain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='Radial strain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='Volumetric strain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='010 StrainGeomechanical risks of CO2 storage 푝푡 = 2휅푇푓퐺 1 + 휈 1 − 2휈 The obtained stress components (23) allow to calculate the stress intensity 푇 and mean stress 휎 under the applied tectonic stress 휎푡 11 and 휎푡 22 (see expressions in Appendix A below Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (47) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (18)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Parameters 푇 and 휎 are substituted into limiting condition (17) and assuming that 휎푡 22 = 푚휎푡 11 we find the critical horizontal stress 휎푡 11(푐푟), at which the horizontal layer turns into inelastic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Next, assuming that the localization of deformations is formed at the stress 휎푡 11(푐푟), consider expression (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Substituting all the known parameters and the angle of fault inclination 휓, we find the corresponding dilatancy coefficient Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Evaluation of rock permeability in the damage zone The calculated dilatancy coefficient allows one to deter- mine the increase in permeability of rock damage zone in the vicinity of the tectonic fault due to plastic deformations along its plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Consider the expression for permeability of the fracture network (Basniev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 1993): 푘푓 = 푚푓훿2 12 , (24) where 훿 is the mean fracture aperture, 푚푓 is the fracture porosity, which is the ratio of the volume of fractures to the total rock volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' These parameters are related with each other by the following expression: 푚푓 = 퐷훿, (25) where 퐷 is the fracture intensity determined experimentally as the ratio of total fracture length to the rock cross-section area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' According to the definition of dilatancy coefficient Λ the following expression holds 휀푝 = ΛΓ푝, (26) where Γ푝 is the intensity of plastic shear deformations (the second invariant of the shear plastic deformation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We assume that the ratio of volume of fractures opened due to rock dilatancy to the geometric volume of the rock is equal to an increase in the rock volume due to inelastic deformations 휀푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Therefore, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (26), we can formulate the following expression for 푚푓: 푚푓 = ΛΓ푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (27) Substituting (27) into (24) and using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (25) we find 푘푓 = (휀푝)3 12퐷2 = (ΛΓ푝)3 12퐷2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (28) Expression (28) allows one to determine an increase in the permeability of near fault damage zone in the presence of plastic deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Note that all the parameters required to use the obtained expression are determined via standard laboratory measurements of rock mechanical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The permeability alteration model described above is embedded into the framework of coupled hydro-geomechanical model (Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Plastic deformations due to change in stress state of the rock during CO2 injection are calculated directly using mechanical simulator FLAC3D with the help of the relation: 휀푝 = 휀푣 − 휀푒, 휀푒 = Δ휎′ 퐾 , where 휀푒 are elastic deformations, 휀푣 is the total volumetric deformation calculated using FLAC3D, Δ휎′ is the change in mean effective stress between the current and initial rock state, and 퐾 is the bulk modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The fracture intensity 퐷 varies between 10 and 50 m−1 (Golf-Rakht, 1986), and in the simulations shown below we assume 퐷 = 30 m−1, which corresponds to sandstone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Effect of shear slip on permeability of fractures closed on natural asperities Modeling of a rock as elastoplastic medium with internal friction and dilatancy allows determining deconsolidation of the fault zone due to shear deformations as described above (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (28)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' This approach can be used to describe the alteration of the fault dynamic influence zone (damage zone) containing microfractures, while it does not allow calculating the aperture of the main crack in the fault core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Major fracture opening due to shear deformations is a result of applied shear stress being larger as compared to the friction force and resistant force of interaction of natural asperities as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Figure 5: Shear slip along the fracture plane, after Barton and Choubey (1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In the framework of Barton-Bandis model (Barton and Choubey, 1977), fracture aperture due to slip 퐸푑 is expressed as follows: 퐸푑 = 푈푠 ⋅ tg(푑푚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (29) where 푈푠 is the slip displacement, and 푑푚 is the dynamic angle of dilatancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Instead of the tangent of 푑푚, we utilize the dilatancy coefficient Λ = tg(푑푚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Laboratory experiments on rock samples show that the dilatancy angle depends on the normal stress, limiting shear stress, friction coefficient at the walls and residual friction angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Zhigulskii and Tikhotskii (2020) show that 3D ge- omechanical modeling allows evaluating mechanical and Preprint submitted to Journal of Natural Gas Science and Engineering Page 8 of 26 a) T=0 b) T≠0Geomechanical risks of CO2 storage hydraulic fracture aperture based on the rock stress state parameters and shear deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Consider the plane problem of linear fracture in an elastic orthotropic medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The fracture of the length 2푎 is oriented along the principal axes of anisotropy and is loaded at infinity with horizontal 푝1, vertical 푝3 and tangential 휏 stresses (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Fracture walls are confined with the stress 푝3 and interact according to dry friction law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Friction coefficient in static state 훼푢푝 is maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' If the shear stress increases up to its critical value 휏푢푝 = 훼푢푝푝3 + 푐, where 푐 is the cohesion, then the fracture walls start to move and friction coefficient drops to the value 훼푑푤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' As a result, the shear stress decreases to the value 휏푑푤 = 훼푑푤푝3 and under the applied stress Δ휏 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 6b) Δ휏 = 휏푢푝 − 휏푑푤 (30) the fracture walls are displaced by 푈푠 = 푢1(푥1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Figure 6: Plane rock domain containing fracture (a) with the walls interacting according to dry friction law (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We define the shear deformation following the study Garagash and Osiptsov (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Equilibrium conditions are formulated as follows 휎11,1 + 휎13,3 = 0, 휎13,1 + 휎33,3 = 0 (31) Consider the function of stresses 퐹 satisfying the follow- ing conditions 휎11 = 퐹,33, 휎33 = 퐹,11, 휎13 = −퐹,13 (32) Under the conditions (32), equilibrium conditions (31) are satisfied identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Strain compatibility condition is formulated as follows 휀11,33 + 휀33,11 = 2휀13,13 (33) The condition (33) can be satisfied by using the consti- tutive relations for transversal isotropic body at plane strain state 휀22 = 0: 휎11 = 퐶11휀11 + 퐶13휀33, 휎33 = 퐶13휀11 + 퐶33휀33, (34) 휎13 = 퐶44휀13 The stiffness tensor components for isotropic medium have the following form: 퐶11 = 퐶33 = 4퐺 1 − 휈 1 − 2휈 , 퐶12 = 퐶13 = 2퐺 1 − 2휈 , 퐶44 = 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Solving (34) with respect to deformations we obtain the following expressions 휀11 = 푆11휎11 + 푆13휎33, 휀13 = 푆44휎13, (35) 휀33 = 푆33휎33 + 푆13휎11, where 푆11 = 퐶33Δ−1 휀 , 푆33 = 퐶11Δ−1 휀 , 푆13 = −퐶13Δ−1 휀 , 푆44 = 퐶−1 44 , Δ휀 = 퐶11퐶33 − 퐶2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (36) Substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (35) into conditions (33) we obtain 푆11퐹,3333 + (2푆13 + 푆44)퐹,1133 + 푆33퐹,1111 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (37) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (37) can be solved using Fourier transformation (Nowacki, 1975), and the following expression for displace- ment along the fracture axis at |푥1| ≤ 푎 is obtained (Gara- gash and Osiptsov (2021)): 푢1(푥1, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='5푎Δ휏(푘1 + 푘2)푆11 √ 1 − (푥1∕푎)2, (38) 푢3(푥1, 0) = −Δ휏 (√ 푆11푆33 + 푆13 ) 푥1, where 푘2 1,2 = 2푆13+푆44 ± √ (2푆13+푆44)2 − 4푆33푆11 2푆11 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Finally substituting displacement 푢1(푥1, 0) into expres- sion (29) we find the fracture aperture 퐸푑: 퐸푑(푥1) = 푢1(푥1, 0) ⋅ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (39) In the mechanical model, we utilize 퐸∗ 푑 value corre- sponding to the averaged displacement profile 푢1(푥1, 0) along the fracture 푥1 ∈ [−푎, 푎]: 퐸∗ 푑 = 휋 4 푎2Δ휏(푘1 + 푘2)푆11Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (40) By using the fracture aperture determined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (40) we calculate the permeability of the main fracture in the fault core closed on natural asperities as follows: 푘푐 = 푤2 푐 12 , (41) where 푤푐 is the hydraulic aperture of the main fracture determined according to Barton and Choubey (1977) as follows: 푤푐 = 퐸2 푑 JRC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='5 , (42) where 퐸푑 is in mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (42), JRC is the joint roughness coefficient determined in the laboratory experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We take JRC = 4 in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' As it is reported in (Barton and Choubey, 1977), JRC varies in the range between 1 and 20 The model of fracture permeability closed on natural as- perities is embedded into the coupling procedure described in Section 2 for the description of the activation of the Preprint submitted to Journal of Natural Gas Science and Engineering Page 9 of 26 a) b) 木T 2 up X3 △T X1 2a Ui P1Geomechanical risks of CO2 storage fault core as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' If the deformations in the mesh cells containing fracture core are pure elastic, and the shear stress 휏푛 on the fault plane reaches the critical value: 휏푛 ≥ 훼푢푝|휎′ 푛| + 푐, (43) where the shear stress 휏푛 and normal effective stress 휎′ 푛 on the fault plane with inclination angle 휓 provided plane strain conditions (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 7) are given by 휏푛 = (휎푧푧 − 휎푥푥) sin 휓 cos 휓 + 휎푥푧 cos 2휓, 휎′ 푛 = 휎′ 푥푥 cos2 휓 + 휎′ 푧푧 sin2 휓 + 휎푥푧 sin 2휓, 휎′ 푥푥 = 휎푥푥 + 푝, 휎′ 푧푧 = 휎푧푧 + 푝, then the fracture aperture 퐸∗ 푑 is calculated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Stress difference Δ휏 is determined based on the stress state evaluated by FLAC3D: Δ휏 = 휏푛 − 훼푑푤|휎′ 푛|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (44) We set 훼푢푝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1, 훼푑푤 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='05, 푐 = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (43), (44) during simulations (see typical values of the friction angle and cohesion in the fault zone in (Ikari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Treffeisen, 2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' For estimation of the prefactor before Δ휏 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (40), we assume that: (i) the fracture length 2푎 equals the height of the mesh cell, which is close to 10 m, so that 푎 ∼ 5 m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (ii) elastic modulus of the transversal isotropic body is given by Bessmertnykh and Dontsov (2018): 퐶11 = 20 GPa, 퐶12 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='8 GPa, 퐶13 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='6 GPa, 퐶33 = 13 GPa, 퐶44 = 3 GPa;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (iii) the dilatancy coefficient is set to Λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' As a result, we obtain 퐸∗ 푑 ≅ 10−9Δ휏, where Δ휏 is in Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Note that the dilatancy coefficient Λ varies in between 0 and 1 (Alejano and Alonso, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Figure 7: Stress components 휏푛 and 휎푛 at the fault plane inclined by angle 휓 to the vertical direction under plane stress conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' If plastic deformations are observed, then the mechanical fracture aperture 퐸푑 is calculated using dilatancy and shear deformations as follows: 퐸푑 = Λ훾푑푥 where 훾 is the intensity of shear deformations (second in- variant of shear stress tensor), and 푑푥 is the mesh cell length in direction perpendicular to the fault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Results and discussion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Verification of coupled hydro-geomechanical model In the current section, we verify the implementation of the in-house algorithm performing coupling between reser- voir simulator MUFITS and mechanical simulator FLAC3D via data transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Both simulators have been verified previ- ously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Results of the benchmark tests of MUFITS are given in papers (Afanasyev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' De Lucia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Afanasyev, 2017) and are available at the web-site of the simulator (Afanasyev, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' At the same time, various tests of the commercial simulator FLAC3D are provided in the manual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We consider the transient fluid flow to a vertical well fully penetrating the infinite-acting aquifer with thickness ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We introduce the cylindrical coordinate system (푟, 휃, 푧) with an origin located at the bottom of the perforation interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 8 shows the schematic representation of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Figure 8: Schematic picture of a vertical production well fully penetrating the infinite-acting aquifer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Initially, the pore fluid pressure 푝0 is uniform, and the reservoir is under isotropic stress 휎0 푧푧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The vertical well produces water at a constant rate 푞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The elastic porous medium is taken as homogeneous and isotropic with drained bulk modulus 퐾, shear modulus 퐺, Biot coefficient 훼, Biot modulus 푀, porosity 휙, permeability 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Pore fluid (water) is described by the viscosity 휇 and bulk modulus 퐾푓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We consider an incompressible solid constituent 훼 = 1 so that 푀 = 퐾푓∕휙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The vertical stress is assumed constant during the water production 휎푧푧(푟, 푡) = 휎0 푧푧, and horizontal strains, 휀푟푟, 휀휃휃, are negligibly small as compared to 휀푧푧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The solution to the formulated problem in terms of the spatial-temporal parameters (pore fluid pressure 푝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' stress tensor components 휎푟푟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 휎휃휃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 휎푧푧,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' and vertical displacement 푢푧) can be derived analytically and is formulated as follows: 푝 = 푝0 − 푞휇 4휋푘ℎ퐸1 ( 푟2 4푐푡 ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 휎푟푟 = 휎휃휃 = 휎0 푧푧 + 푞훼퐺휇 2휋푘ℎ훼1 퐸1 ( 푟2 4푐푡 ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 휎푧푧 = 휎0 푧푧,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Preprint submitted to Journal of Natural Gas Science and Engineering Page 10 of 26 Z 山 X m xzGeomechanical risks of CO2 storage Parameter Value,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' unit 퐾 20 GPa 퐺 10 GPa 퐾푓 2 GPa 휇 1 cP 휙 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1 푘 1 mD 훼 1 푞 4 m3/day 푝0 100 bar 휎0 푧푧 100 bar Table 1 Parameters of the vertical well model considered in the numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 푢푧 = − 푧훼푞휇 4휋푘ℎ훼1 퐸1 ( 푟2 4푐푡 ) , (45) where 훼1 = 퐾 + 4퐺∕3, 푐 = 푘∕(휇푆) is the diffusion coefficient, 푆 = 1∕푀 + 훼2∕훼1 is the storage coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We solve the formulated problem numerically using two approaches: (i) developed coupled hydro-geomechanical model based on MUFITS and FLAC3D and (ii) FLAC3D (FLAC3D can simulate single phase flow of slightly com- pressible liquid in addition to the mechanical calculations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Subsequently, model (ii) is applied to verify the imple- mentation of porosity and permeability alteration in the hydrodynamical model according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (12), (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Table 1 outlines the values of model parameters used in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Note that it is necessary to put the adjusted fluid modulus into the hydrodynamic model 퐾푎 푓 = 휙∕ (휙∕퐾푓 + 1∕훼1 ) in order to preserve the real diffusivity (in the expression, we take into account the Biot coefficient value 훼 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We model the water production within 60 days, and the outer boundary of the reservoir located in the numerical models at a distance of 1 km from the producer is not reached by the pressure wave during the simulation so that the condition of the infinite-acting reservoir is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 9 compares two numerical solutions computed by MUFITS+FLAC3D and FLAC3D with the analytical so- lution given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We look at the distributions of pressure, vertical displacement, radial and vertical stresses over the coordinate interval 푟 ≤ 500 m at different time moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Since the vertical stress is constant over time, we depict the numerical solution for this parameter at the end of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' One can conclude that the outcome of the proposed coupled hydro-geomechanical model based on MUFITS and FLAC3D matches acceptably with the analytical solution of the problem, and we make similar inference regarding the numerical model built on FLAC3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In the numerical experiment shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 9, we check the data transfer from MUFITS to FLAC3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' However, we should also verify the reverse data flow between simulators (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', from FLAC3D to MUFITS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' For that purpose, we amend the numerical models by adding the alteration of porosity and permeability in the hydrodynamical part after each mechanical calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We embed the following rela- tions for porosity and permeability: 휙 = 1 − (1 − 휙0)푒−50⋅휀푣, 푘 = 푘0(휙∕휙0)8, where 휙0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1, 푘 = 1 mD, so that we modify artificially Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (12) to increase the impact of volumetric deformations on porosity, while the value of the power-law exponent in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (13) is within the typical range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Moreover, in the current numerical experiment, it is not required to adjust the fluid modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' As a result, for comparison purposes, we can demonstrate the offset of the numerical solution from an analytical one as described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (45), which corresponds to the storage coefficient 푆 = 휙∕퐾푓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 10 shows the obtained results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We demonstrate the distributions of pressure, vertical displacement, radial stress, and permeability over the coordinate interval 푟 ≤ 100 m at different time instants (permeability evolution is shown within the area 푟 ∈ [0, 500] m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The numerical solutions deviate from the analytical ones after a couple of days of water production, the discrepancy is observed at 푡 = 5 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We would like to stress that the analytical curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 10 are not the solution to the problem under consideration, and they correspond to the water flow in an incompressible porous medium with constant porosity 휙0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1 and perme- ability 푘0 = 1 mD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Moreover, one can observe a satisfac- tory match between the results of simulations obtained via MUFITS+FLAC3D and FLAC3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In other words, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 10 confirms the correct implementation of the data transfer from FLAC3D to MUFITS performed via the porosity and permeability modification in the hydrodynamical model im- plemented in MUFITS based on deformations computed by FLAC3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Coupled hydro-geomechanical modeling of CO2 sequestration in an aquifer on the example of the synthetic formation case The current section presents the simulation results of CO2 injection into the target aquifer intersected by the tectonic fault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We construct a synthetic model of two- dimensional multilayered formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' During the interpre- tation of the calculations, we focus on the development of undesired mechanical processes, namely, slip along the fault plane resulting in the crack opening on the asperities, the plastic deformations in the fault zone, target aquifer, and caprock, as well as carbon dioxide leakage along the fault zone towards the overlying collector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Model description Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 11 shows the schematic representation of the utilized synthetic model of a two-dimensional multilayered reservoir with the following geometrical parameters: the lateral size of the reservoir is 1500 m, while its height equals 600 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Two cases of the reservoir depth are considered: the upper bound locates at a depth of (i) 600 m and (ii) 1700 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Carbon dioxide is injected into the target aquifer of thickness 100 m surrounded by the low permeable caprock and basement layers, each of 150 m height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' These three layers are intersected by the tectonic fault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The fault zone Preprint submitted to Journal of Natural Gas Science and Engineering Page 11 of 26 Geomechanical risks of CO2 storage Figure 9: Results of the analytical and numerical modeling of transient fluid flow to a vertical well fully penetrating the infinite- acting poroelastic reservoir in terms of pressure (a), vertical displacement (b), radial stress (c) and vertical stress (d) distributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The analytical solution is shown by solid lines, while the numerical solutions are given by markers, MUFITS+FLAC3D by crosses and FLAC3D by circles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' the solutions correspond to the following time instants: 푡 = {1, 5, 15, 30, 60} day(s);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' zoomed domain in the vicinity of the well 푟 ∈ [0, 100] m is shown in plots (a) – (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' thickness is 20 m, the inclination angle relative to the vertical direction is 15◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The distance between the left edge of the formation and the fault at a depth corresponding to the middle of the storage aquifer is 550 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The fault consists of the damage zone and core, the thickness of latter one is 1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The upper and basal aquifers are placed above and below the caprock and basement layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The injector has a vertical completion coinciding with the left border of the reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' CO2 injection is carried out through the perforations located along the intervals: (i) 푧 ∈ [−910, −890] m and (ii) 푧 ∈ [−1910, −1890] m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Before CO2 injection, the formation is assumed to be water-saturated, the pressure distribution is hydrostatic (in the general case, it is computed from the phases distribution and gravitational-capillary equilibrium), and the tempera- ture field corresponds to the geothermal gradient of 25 C/km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' FLAC3D computes the in-situ mechanical state of the reservoir using the prescribed pressure and temperature at the initial time instant and the geological model (distri- butions of density and elasticity modulus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The calculated initial displacements are set to zero, so that the subsequent deformations appearing due to CO2 injection are measured from the in-situ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The vertical well injects carbon diox- ide at constant bottomhole pressure: (i) 150 bar and (ii) 300 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We choose the bottomhole pressure values relying on the minimal principal stresses 휎min observed at a depth of perforations in the first (i) and second (ii) cases as follows: bottomhole pressure should be lower than 휎min (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', by 10 %), to prevent the initiation of hydraulic fractures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The top and left edges of the reservoir are impermeable (solid black lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' At the bottom and right boundaries, we specify a constant pore pressure equal to the initial one (dashed black lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' At the left and right edges of the formation, we fix zero displacements along x-axis 푢푥 = 0, while the displacements along x and z-axis are prohibited at the bottom boundary 푢푥 = 푢푧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In addition, we fix the displacement along z-axis at the right border 푢푧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' At the Preprint submitted to Journal of Natural Gas Science and Engineering Page 12 of 26 a) 100 c) 75 75 80- 80 - 90- 100- 85 Orr, bar 85- 90- bar 90 bar 80 Orr, 95- 80 06- 100 0 20 40 60 80 100 70- 70 区 r, m 95 X 60 0 20 40 60 80 100 100 60- r, m 0 100 200 300 400 500 0 100 200 300 400 500 r, m r, m b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0 95 analytical t= 96- X MUFITS+FLAC3D 1 d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='2 0 FLAC3D 5 d X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0 97 15 d ww 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content="2 bar 98- 30 d Ozz' 60 d 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='6- 99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='8- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='8 X 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0 0 20 40 60 80 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0 r, m 101 0 100 200 300 400 500 0 100 200 300 400 500 r, m r, mGeomechanical risks of CO2 storage Figure 10: Results of the numerical modeling of transient axisymmetric fluid flow to a vertical well fully penetrating the infinite-acting reservoir accounting for the porosity and permeability alterations based on the mechanical calculations: pressure (a), vertical displacement (b), radial stress (c) and permeability (d) distributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The numerical solutions are given by markers, MUFITS+FLAC3D by crosses and FLAC3D by circles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' analytical solution corresponding to the pore fluid flow in the incompressible porous medium with the constant porosity and permeability (dashed lines);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' we demonstrate the solutions along the spatial domain 푟 ∈ [0, 100] m (푟 ∈ [0, 500] m in plot d) at the following time instants: 푡 = {1, 5, 15, 30, 60} day(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Figure 11: Synthetic reservoir model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' solid and dashed black lines denote impermeable and constant pressure boundaries, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' the red arrow marks the perforation interval through which CO2 is injected into the target aquifer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' by red and purple colors we show core and damage zones in the fault domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' top boundary, we apply constant loading 휎푧푧 corresponding to the lithostatic pressure created by a layer of thickness (i) 600 m and (ii) 1700 m with a density 2400 kg/m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Table 2 provides the model parameters: mechanical properties: Young’s modulus 퐸, Poisson ratio 휈, cohesion 푐, angle of internal friction 휃, dila- tancy Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' porosity 휙, permeability 푘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' rock density 휌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We set the specific heat capacity of rock 퐶푟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='81 kJ / (kg ⋅ K) and heat conductivity of saturated porous medium 휆 = 3 W / (m ⋅ K) for the entire domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Pore water salinity is neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Note that the chosen permeability values of the dam- age zone and fault core are consistent with the experi- mental measurements and estimations provided in papers Preprint submitted to Journal of Natural Gas Science and Engineering Page 13 of 26 a) 100- C 70 t = 75- 1 d X4 90 5 d 区 80 15 d bar bar X 80 30 d X 85 Orr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' X p 60 d X 90 由 X 70 区 analytical Kf X 区 这 X 95 区 X MUFITS+FLAC3D 60 8 0 FLAC3D 100 0 20 40 60 80 100 0 20 40 60 80 100 r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' m r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' m b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' d) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='00 & & α & 区 & X & 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='95 区 区 & & R X 区 & & 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='90 & 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='4 文 & X ww 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='85 D m & & 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='6 zn K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='80 & 文 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='8 - & 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='70 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='65 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='2 0 20 40 60 80 100 0 100 200 300 400 500 r, m r, mCO2 X Upper aquifer 100 m Caprock 150 m 550 m Target aquifer 100 m :15° Basement 150 m 20 m Basal aguifer 100 m 1500 mGeomechanical risks of CO2 storage Layer 휙 푘, mD 퐸, GPa 휈 휌, kg/m3 푐, MPa 휃 Λ Upper aquifer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1 10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='2 2400 Caprock 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='01 10−4 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='15 2400 4 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='2 Target aquifer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1 100 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='2 2400 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='6 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='4 Basement 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='01 10−4 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='3 2600 Basal aquifer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='3 2600 Fault (damage zone) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1 2400 Fault (core) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1 10−3 2400 Table 2 Mechanical and flow properties of the synthetic reservoir model depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The values of cohesion, angle of internal friction, and dilatancy corresponding to the upper aquifer, basement, and basal aquifer are absent, since these layers are considered as elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We set the values of the elastic modulus and strength parameters in the fault zone in such a way as to reproduce its complex structure, and one can find the description of this procedure in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (Faulkner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Wibberley and Shimamoto, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Scibek, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The authors of these studies conclude that the fault core permeability is typically lower than that of the damage zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' As a result, the fault permeability along its plane is larger than the permeability in the normal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 12 shows the utilized relative permeabilities for the gas (black line) and liquid (red line) phases, as well as the capillary pressure curve (blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' These parameters depend on the liquid phase saturation 푠푙 according to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (7) with 푠푙푟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='3, 푠푔푟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='05, 휆푙 = 휆푐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='457, 푃푐0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1961 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Figure 12: Relative permeabilities and capillary pressure curves embedded into the hydrodynamic reservoir model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We specify that the target aquifer and caprock including the intersected fault zone are described by the elastoplas- tic rheological model based on the Drucker-Prager yield condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The remaining layers are elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The choice is motivated by the aim to track the development of plastic deformations in the regions ensuring the loss of integrity of the carbon dioxide storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The spatial meshes in the reservoir simulator MUFITS and mechanical simulator FLAC3D are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The di- mensions of mesh cells out of the fault zone are Δ푥 = 20 m, Δ푧 = 10 m (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 13) with slightly variation towards the fault, where the columns of cells become parallel to the fault plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We apply the grid refinement in the fault zone, which allows reproducing its complex structure including the core surrounded by the damage zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Seven columns of cells are used to approximated the fault zone, and their thickness de- creases towards the fault center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The central column denotes the fault core and contains the main crack, which is initially healed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The remaining 6 columns (3 to the left and right of the fault core) describe the damage zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Elastic modules (bulk modulus 퐾 and shear modulus 퐺), cohesion 푐, and angle of internal friction 휃 decrease towards the fault core in accordance with the studies Gudmundsson (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Faulkner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Treffeisen (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We assume that the values of these parameters at the fault core comprise 70% of those corresponding of the host rock at the considered depth (see typical values of the contrast in the mechanical properties between the host rock and fault core in Holdsworth (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Collettini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Treffeisen (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The trend for variation of the dilatancy coefficient Λ is similar, but it increases towards the fault center so that its values at the fault core are 30% larger than that in the host rock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Figure 13: Reservoir spatial discretization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' the mesh structure in the fault zone is zoomed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' colors highlight different layers and fault zone (see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' CO2 injection is simulated for 30 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' During the first 5 years, we compute the mechanical equilibrium state every 6 months using FLAC3D simulator and reservoir porosity and permeability are updated according to the current stress state and deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' After that period, FLAC3D simulator is called once a year for 25 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Modeling results Target aquifer at 950m depth We start with discussion of the results of pure hydrody- namical modeling of CO2 injection using MUFITS simula- tor (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' After 30 years of injection, the CO2 plume Preprint submitted to Journal of Natural Gas Science and Engineering Page 14 of 26 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='95 KRGAS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' KRLIQ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='85 PCGL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='3 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='95 Liquid saturation-600 650 700 750 800 850 900 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='950 1000 1050 1100 1150 1200 500 -400 -300 -200 -100 0 100 200300 400 500 600 700 800 9001000 x,mGeomechanical risks of CO2 storage reaches the fault zone and crosses it (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 14a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' On the right side of the fault, CO2 flows along the upper part of the target aquifer and passes 400 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Carbon dioxide also flows along the tectonic fault where the CO2 plume uplifts at a distance of 50 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 14b, one can notice that the storage aquifer leftward to the fault zone contains the pressure plume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Increase in pore pressure (the difference in pore pressure values at the final and initial time instants) is homogeneous in the target aquifer due to the high contrast in permeability values corresponding to the CO2 storage do- main and surrounding layers, where the pressure disturbance gradually decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' On the left hand of the fault, the pore pressure increase is observed in the caprock and basement layers only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Consequently, according to the hydrodynamic simulation, there is no leakage of CO2 from the target layer to the upper aquifer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Figure 14: Results of hydrodynamical modeling of CO2 injection into the target aquifer with the upper boundary located at a depth of 600 m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' plots (a) and (b) show the gas saturation and pore pressure increment (as compared to the initial hydrostatic distribution) fields at the end of the simulation period (30 years).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In the framework of coupled simulations we consider two cases of the initial mechanical state: 휎푥푥 = 휎푦푦 (no tectonic stresses) and 휎푦푦∕휎푥푥 ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='7 at the depth of the target aquifer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We begin with the case of no tectonic stresses, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 15 presents the results of simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Carbon dioxide flows along the fault, and the CO2 plume reaches the upper aquifer as it can be noted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 15a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The opening of the main crack facilitates this behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' During CO2 injection, the slip along the fault plane in an elastic mode occurs, and the fracture opens at natural asperities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' After 30 years of CO2 sequestration, the main crack opens along its entire length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' However, the fracture aperture at the depth interval corresponding to the caprock and storage aquifer is smaller as compared to that at the basement layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Plastic deforma- tions do not develop in the reservoir so that the permeability increase in the fault zone is observed only in the direction parallel to the fault plane due to opening of the main fracture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The appearance of the conduit leads to CO2 flow along the fault to the upper aquifer and the carbon dioxide leakage out of the disposal zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Moreover, rightward to the fault zone, CO2 flow occurs along the shorter distance compared to that obtained using pure hydrodynamical modeling (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 14a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 15d demonstrates the mass of the injected CO2 in the coupled model (red line) and hydrodynamical model (blue line), and the former one is higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The pore pressure increment shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 15b is similar to that obtained using hydrodynamical model (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 14b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' However, the pressure increase on the right side of the fault in the caprock and basement layers is more pronounced in the coupled model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Volumetric strain is maximal in the target aquifer leftward to the fault and declines towards the upper and basal aquifers in the caprock and basement layers (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 15c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Next, we discuss the results of the coupled hydro- geomechanical modeling of CO2 injection into the target aquifer with pronounced tectonic stresses at the initial state as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The ratio 휎푦푦∕휎푥푥 ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='7 is set to observe the development of plastic deformations in the fault zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' After 30 years of carbon dioxide injection, the main crack opens along the entire length, and the fracture aper- ture decreases towards the basement layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Intense plastic deformations develop in the fault zone at the depth of the target aquifer so that the natural fractures open in the damage zone of the tectonic fault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Thereby, the permeabilities of the fault along and perpendicular to its plane are increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' As a result, we observe a significant CO2 leakage into the upper aquifer and CO2 flow in the target aquifer on the right side of the fault (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 16a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 16d, it is clear that the injected mass of CO2 in the coupled hydro-geomechanical model is much higher as compared to the hydrodynamic simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The distributions of the pore pressure increment and volumetric strain (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 16b and c) are similar to that obtained in the case of no tectonic stresses (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 15) except for the domain of the upper aquifer where both parameters grow due to the leakage of carbon dioxide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Additionally, note the significant volumetric strain and pore pressure in the fault zone at a depth of the caprock layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Target aquifer at 2000m depth Now we discuss the results of the modeling of CO2 injection into the target aquifer of the formation with the upper boundary located at the depth of 1700 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We begin with the hydrodynamic simulation (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' At the end of the simulation period (30 years), the CO2 plume reaches the fault zone and crosses it (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 17a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' During the flow rightward to the fault, CO2 passes 900 m reaching approximately the right boundary of the reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We also observe CO2 flow along the fault zone, and carbon dioxide rises at a distance of 150 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The shape of the pore pressure plume demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 17b is similar to that obtained in the previous case of the reservoir depth of 950 m (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 14b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The difference is that the pressure increase with respect to the initial distribution is higher due to larger bottomhole pressure value (300 bar versus 150 bar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Thus, the hydrody- namic computation demonstrates that the CO2 plume almost reaches the upper aquifer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Preprint submitted to Journal of Natural Gas Science and Engineering Page 15 of 26 a) 600 650 700 750 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e-01 800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='4 N-950 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='3 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='2 1050 1100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1 1150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e+00 1200 500 400-300-200-100 0 100 200300 400 500 600 700 800 9001000 x,m b) 600 650 Ap, bar 700 750 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='5e+01 800 50 40 N-950 30 1000 20 1050 1100 10 1150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e+00 120Q 500 400-300 200 100 0 100 200300 400 500 600 700 800 9001000 x,mGeomechanical risks of CO2 storage Figure 15: Results of coupled hydro-geomechanical modeling of CO2 injection into the target aquifer with the upper boundary located at a depth of 600 m in the absence of tectonic stresses (휎푥푥 = 휎푦푦);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' plots (a) – (c) show the gas saturation, pore pressure increment (as compared to the initial hydrostatic distribution), and volumetric strain fields at the end of the simulation period (30 years), respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' plot (d) depicts the dynamics of injected mass of carbon dioxide in pure hydrodynamical model (red curve) and coupled model (blue curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Figure 16: Results of coupled hydro-geomechanical modeling of CO2 injection into the target aquifer with the upper boundary located at a depth of 600 m at the pronounced tectonic stresses (휎푦푦∕휎푥푥 ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='7 in the target aquifer);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' plots (a) – (c) show the gas saturation, pore pressure increment (compared to the initial hydrostatic distribution), and volumetric strain fields at the end of the simulation period (30 years), respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' plot (d) depicts the dynamics of injected mass of carbon dioxide in pure hydrodynamical model (red curve) and coupled model (blue curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Let us consider the results of the coupled simulations in the absence of tectonic stress at the initial mechanical state 휎푥푥 = 휎푦푦 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Plastic deformations do not develop in the reservoir, and the main crack opens along the entire length in the elastic mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The formed conduit contributes to the leakage of carbon dioxide into the upper aquifer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The CO2 plume crosses the fault zone and spreads along the target layer on the right side of the fault over a shorter distance as compared to that obtained using pure hydrodynamical calculations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 18a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 18d compares the mass of injected CO2 in the coupled and hydrodynamical simulations, and the former one is higher due to the major crack opening on asperities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The shape of the pressure plume shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 18b is similar to that obtained in the previous configuration of the formation with no tectonic stresses (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 15b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The target aquifer exhibits large values of the volumetric strain (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 18c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We also observe the reduced volumetric deformations in the vicinity of the injector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We attribute this to the temperature effect since the injected carbon dioxide is colder as compared to the reservoir water inside the target aquifer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In the present case, CO2 cools the reservoir in the vicinity of injector by 40 degrees, while in the previous case (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 15c), we do not observe noticeable influence of the non-isothermal flow on the mechanical equilibrium state of the formation due to small difference in between CO2 and pore fluid temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Preprint submitted to Journal of Natural Gas Science and Engineering Page 16 of 26 a) 600 c) 600 650 650 & 700 700 750 7.' metadata={'source': 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+page_content='0e+3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e+3 1200 500-400-300-200-1000 1002003004005006007008009001000 x,mGeomechanical risks of CO2 storage Figure 17: Results of hydrodynamical modeling of CO2 injection into the target aquifer with the upper boundary located at the depth of 1700 m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' plots (a) and (b) show the gas saturation and pore pressure increment (compared to the initial hydrostatic distribution) fields at the end of the simulation period (30 years), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 19 shows the results of coupled hydro-geomechanical modeling of CO2 injection into the formation in the presence of pronounced tectonic stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The ratio of principal stresses at the target layer 휎푦푦∕휎푥푥 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='8 is set to facilitate the development of plastic deformations in the fault zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Note the plastic deformations are formed at a lower value of the ratio 휎푦푦∕휎푥푥 as compared to the prevoius case of target aquifer located at the depth of 950 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' After 30 years of carbon dioxide injection, the main fracture opens along the entire length of the fault zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We obtained smaller values of the crack aperture at the depth interval corresponding to the bottom segment of the storage aquifer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Moreover, we observe the development of the plastic deformations at the fault zone at the caprock layer and target aquifer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The opened major crack in the core and natural fractures in the damage zone of the fault contribute to the CO2 leakage into the upper aquifer and CO2 flow towards the right border of the reservoir (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 19a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The latter effect results in the larger volume of CO2 plume rightward to the fault as compared to that obtained using the hydrodynamic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The mass of injected CO2 is substantially larger in the case of the coupled model (red curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 19d) as compared to that obtained using the hydrodynamic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The pore pressure in the plume differs from that obtained in the case of no tectonic stresses by a tangible pressure increase in the upper aquifer due to the carbon dioxide leakage (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 19b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Volumetric deformation is maximal in the fault zone at the depth of the caprock layer (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 19c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Still they are large in the target aquifer on the left side of the fault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The reduced values of the volumetric strain are attributed to the thermal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Coupled hydro-geomechanical modeling of CO2 sequestration in an aquifer of the real formation In the current section, we show the results the modeling of CO2 injection into an aquifer using the proposed coupled hydro-geomechanical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We consider a slice of the reservoir sector that contains a tectonic fault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Similar to Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='2, the model is two-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' For the model construction we use the field data collected from well log- ging, well test, seismic survey, and laboratory experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Model description Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 20 shows the schematic representation of the for- mation under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Here, we illustrate its structure, geometrical characteristics, and the tectonic fault placement relative to the injector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The reservoir has the length and height of 1500 m and 1350 m, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The upper boundary of the formation is located at the depth of 1350 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Carbon dioxide is injected into the upper and lower aquifers through a vertical well located at the left boundary of the reservoir;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' perforations are distributed along the interval 푧 ∈ [−2300, −2100] m except for the 10 m thick caprock layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 20 one can observe a layer of anhydrite and a massive layer of salt above the upper aquifer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' A tectonic fault starts at the salt layer and finishes at the basement crossing the anhydrite layer as well as aquifers and caprock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The fault is almost vertical with an inclination angle of 3◦ with respect to the vertical direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The distance between the injector and the fault equals 360 m at 푧 = −2300 m corresponding to the middle of the lower aquifer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We embed the same fault structure into the coupled model as considered in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='2, namely, the fault has thickness of 20 m, while its core thickness is set to 1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Carbon dioxide is injected into the upper and lower aquifers at fixed bottomhole pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We consider two cases of injection with the following bottomhole pressures: (i) 350 bar (base case) and (ii) 600 bar (upper limit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The boundary conditions in the hydrodynamic and mechanical models are similar to the synthetic reservoir model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The difference is in the constant loading applied at the upper border of the formation: in the current model, 휎푧푧 corresponds to the lithostatic pressure created by a layer of thickness 1350 m with a density 2400 kg/m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We describe the mechanical and flow properties of the formation in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The relative permeabilities and capil- lary pressure curve are the same as provided in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Principal components of the tectonic strain tensor at the initial state are 휀푡 푥푥 = −10−4, 휀푡 푦푦 = −3 ⋅ 10−4 at a depth of the lower aquifer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Using the relations Σ푡 푥푥 = 퐸 (1 − 휈2) ( 휀푡 푥푥+휈휀푡 푦푦 ) , Σ푡 푦푦 = 퐸 (1 − 휈2) ( 휀푡 푦푦+휈휀푡 푥푥 ) , we compute the principal components of the tectonic stress tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In these relations, Young’s modulus 퐸 and Poisson ratio correspond to the lower aquifer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Since we are interested in the tectonic stresses observed at the plane of the examined Preprint submitted to Journal of Natural Gas Science and Engineering Page 17 of 26 1700 a) 1750 S 1800 gas 1850 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e-01 1900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='4 N-2050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='3 2100 2150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='2 2200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1 2250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e+00 2300 500 400-300-200-100 0 100 200 300 400 500 600 700 8009001000 x,m 1700 b) 1750 Ap, bar 1800 1850 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1e+02 1900 80 N-2050 60 2100 40 2150 20 2200 2250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e+00 230Q 500 400-300 200 100 0 100 200300 400 500 600 700 800 9001000 x,mGeomechanical risks of CO2 storage Figure 18: Results of coupled hydro-geomechanical modeling of CO2 injection into the target aquifer with the upper boundary located at a depth of 1700 m with no tectonic stresses (휎푥푥 = 휎푦푦);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' plots (a) – (c) show the gas saturation, pore pressure increment (compared to the initial hydrostatic distribution), and volumetric strain fields at the end of the simulation period (30 years), respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' plot (d) depicts the dynamics of injected mass of carbon dioxide in pure hydrodynamical model (red curve) and coupled model (blue curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Figure 19: The figure presents the results of coupled hydro-geomechanical modeling of CO2 injection into the target aquifer with the upper boundary located at a depth of 1700 m accounting for the tectonic stresses (휎푦푦∕휎푥푥 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='8 in the target aquifer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Panels (a) – (c) show the gas saturation, pore pressure increment (compared to the initial hydrostatic distribution), and volumetric strain fields at the end of the simulation period (30 years).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Panel (d) depicts the dependence of the injected mass of carbon dioxide on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Layer 휙 푘, mD 퐸, GPa 휈 휌, kg/m3 푐, MPa 휃 Λ Salt 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='01 10−4 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='44 2100 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='5 28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1 Anhydrite 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='05 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='26 3000 16.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='01 10−4 38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='3 2680 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1 29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1 Lower aquifer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='6 34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='27 2610 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='4 38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1 Basement 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='01 10−4 38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='3 2680 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1 29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1 Fault (damage zone) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='6 2600 Fault (core) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1 10−2 2600 Table 3 Mechanical and flow properties of the realistic formation shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 20);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' all layers are governed by the elastoplastic constitutive model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' the distribution of the mechanical properties inside the fault zone is similar to that described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Preprint submitted to Journal of Natural Gas Science and Engineering Page 18 of 26 1700 1700 a) 1750 c) 1750 S Ev 1800 gas 1800 1850 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e-01 1850 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e-03 1900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='6 1900 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e+4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e+4 Ni-2050 60 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='00+4 2100 40 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e+4 2150 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e+4 20 2200 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e+00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e+4 2250 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e+4 2300 1002003004005006007008009001000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e+4 500-400-300-200-1000 x,m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0 23456789101112131415161718192021222324252627282930Geomechanical risks of CO2 storage Figure 20: The slice of the sector of the real formation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' by solid and dashed black lines we show impermeable and constant pressure boundaries, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' the yellow bars mark the perforation intervals through which CO2 is injected into the upper and lower aquifers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' reservoir slice, we utilize the following expressions: 휎푡 푥푥 = Σ푡 푥푥 cos2 휓 + Σ푡 푦푦 sin2 휓, 휎푡 푦푦 = Σ푡 푥푥 sin2 휓 + Σ푡 푦푦 cos2 휓, 휎푡 푥푦 = 1 2 ( Σ푡 푦푦 − Σ푡 푥푥 ) sin 2휓, where 휓 is the angle between the principal x-direction and the slice plane (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In the current case, the angle 휓 equals 15◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Figure 21: Components of the tectonic stress tensor in the coordinate axes 푥′푦′ (휎푡 푖푗) and in the principal axes 푥푦 (Σ푡 푖푗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In the mechanical reservoir model, we describe the rheology of all layers by the elastoplastic model with the Drucker-Prager yield condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The spatial meshes in MU- FITS and FLAC3D are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The cell dimensions are Δ푥 = 20 m and Δ푧 = 10 m in the domain outside of the fault zone, where we utilize the same grid refinement in the fault zone as in the synthetic reservoir model (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='2) to represent its complex structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Carbon dioxide is injected for 30 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Modeling results We start with the results of the base case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Here, carbon dioxide is injected into the upper and lower aquifers at a constant bottomhole pressure of 350 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 22 shows the gas saturation, pore pressure increment, and volumetric strain distributions at the end of the simulation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' After 30 years of CO2 injection, the carbon dioxide plume does not reach the fault zone (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 22a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Its maximum lateral size and height are 250 m and 450 m, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Large pore pressure increase is observed inside the domain occupied by the CO2 plume (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 22b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The pore pressure perturbations extend across the entire thickness of the reser- voir and along the distance of 1 km from the injector in the lateral direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Thus, the pore pressure plume dimensions are much larger as compared to that of CO2 plume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We observe the large values of the volumetric strain in both aquifers and in the lower part of the salt deposit (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 22c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Similar to the synthetic reservoir model (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='2), the reduced values of the volumetric strain near the perforations are attributed to the cooling effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Plastic deformations are not developed in the reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The main fracture does not open so that the condition Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (43) is not satisfied along the entire crack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In the current case, permeability can increase in the for- mation due to the volumetric deformations only according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Since deformations are relatively small, the changes in porosity and permeability are also small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' For example, permeability increase is less than 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Comparing dynamics of the mass of injected carbon dioxide in the case of coupled modeling and hydrodynamic simulation, we obtain a negligible difference between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Next, we move on to the results of the case in which the bottomhole pressure is fixed to 600 bar, which exceeds the minimum principal stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 23 illustrates the distributions of the gas saturation, pore pressure increment, and volumet- ric strain after 30 years of CO2 injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 23a it is clear that the CO2 plume reaches the fault zone, crosses it in both aquifers, and distributes partially along the damage and core zines of the fault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Rightward to the fault, CO2 flows in the bottom part of the upper aquifer, and throughout the entire thickness of the lower aquifer, where the maximum lateral size of the CO2 plume is about 550 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The shape of the pore pressure plume shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 23b is similar to the previous case, in which bottomhole pressure equals 350 bar (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 22b): pore pressure diffuses over the entire reservoir thickness and along the domain of 1 km length in horizontal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Moreover, the volumetric strain field is qualitatively similar to that obtained in the previous case (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 22c), while the deformations are larger by an about an order of magnitude due to the higher pore pressure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 23c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Plastic deformations do not develop in the fault zone,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' but we observe them in the vicinity of the perforation interval located in the upper aquifer and in the bottom part of the salt deposit near the left boundary of the Preprint submitted to Journal of Natural Gas Science and Engineering Page 19 of 26 1400 1500 CO2 salt 1600 1700 1800 anhydrite 1900 三 -2000 upper aquifer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' limestone N-2100 caprock,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' dense limestone 2200 2300 360 m lower aguifer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='limestone 2400 basement,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 2500 2600 dense limestone 2700 0 200 400 600 800 1000 1200 1400 x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='myy 七 yy xx xxGeomechanical risks of CO2 storage Figure 22: Results of coupled hydro-geomechanical modeling of CO2 injection into the upper and lower aquifers at a constant bottomhole pressure of 350 bar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' plots (a) – (c) show the gas saturation, pore pressure increment (compared to the initial hydrostatic distribution), and volumetric strain fields at the end of the simulation period (30 years), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Figure 23: Results of coupled hydro-geomechanical modeling of CO2 injection into the upper and lower aquifers at a constant bottomhole pressure of 600 bar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' plots (a) – (c) show the gas saturation, pore pressure increment (compared to the initial hydrostatic distribution), and volumetric strain fields at the end of the simulation period (30 years), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The former indicate the possible appearance of hydraulic fractures near the injector since the bottomhole pressure exceeds the minimal stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The main crack does not open, and despite the increased value of the bottomhole pressure (600 bar versus 350 bar), we still obtain that the slip along the fault plane does not occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The maximum permeability increase observed in the aquifers is about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='5%, while this value in the damage zone of the fault reaches 4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Due to the improvement of permeability in the target layers, the mass of the injected CO2 is higher in the coupled simu- lation as compared to that obtained using the hydrodynamic simulation (red line compared to the blue line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' However, the difference in the injected mass at the end of the simulation period is insignificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Sensitivity analysis of the coupled hydro-geomechanical model In the current section, we perform the sensitivity anal- ysis of the coupled hydro-geomechanical model by vary- ing mechanical and flow properties of the reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We analyze the fault stability as well as the development of plastic deformations leading to the loss of integrity of the storage domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The simulations are carried out for realistic parameters determining CO2 storage in the aquifers of at the bottomhole pressure of 600 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Figure 24: Dynamics of the CO2 mass injected at a constant bottomhole pressure 600 bar computed via the coupled hydro- geomechanical (red line) and hydrodynamic (blue line) models during the second half of the injection period (15-30 years);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' results of simulations using modified coupled models (see details in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='3) are also shown: with reduced values of the mechanical characteristics in the aquifers and caprock (green line), with strengthened contrast in the mechanical properties between the host rock and the fault core in the fault zone (dashed blue line), with increased permeability in the fault zone (green line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We assume that the mechanical properties of both aquifers and caprock layer between them can vary in certain ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Preprint submitted to Journal of Natural Gas Science and Engineering Page 20 of 26 a) 1400 b) 1400 c) 1400 Ap,bar 43 1500 gas 1500 1500 1600 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e-01 1600 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='5e+02 1600 5.' metadata={'source': 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+page_content='1 N-2100 50 N-2100 0 2200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e+00 2200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e+00 2200 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e-04 2300 2300 2300 2400 2400 2400 2500 2500 2500 2600 2600 2600 2700 2700 2700 0 200 400 600 800 100012001400 0 200 400 600 800 100012001400 0 200400 600800 100012001400 x,m x,m x,mBottomhole pressure 600 bar 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e+5 COUPLEDBASE COUPLED REDUCED 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='5e+4 COUPLFD FAUL COUPLED PERM 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e+4 HYDRO 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='5e+4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e+4 ton 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e+4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='5e+4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e+4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='5e+4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e+4 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Time, yearGeomechanical risks of CO2 storage Layer 퐸, GPa 휈 휌, kg/m3 푐, MPa 휃 Upper aquifer 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='16 2350 13 35 Caprock 28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='25 2580 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='3 26 Lower aquifer 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='23 2360 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='7 36 Table 4 Decreased values of mechanical properties of the aquifers and caprock in the realistic formation model shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='2, we put into FLAC3D simulator their average values (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' However, one can suggest that the minimal values of the elastic modulus and strength parameters can facilitate the undesired mechanical effects related to the tectonic fault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We carry out the coupled simulations using the decreased values of the mechanical parameters outlined in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 24, we compare the dynamics of the injected carbon dioxide mass during the second half of the simulation period obtained by two coupled models: with the average (red line) and reduced (green line) values of the mechanical properties (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We find that in the case of modified properties the injected mass is larger as compared to than obtained in the base case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The larger mass of the injected CO2 is associated predominantly with the larger volumetric strains (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 25b) leading to the improvement of porosity and permeability (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (12), (13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In contrast to the base model, we do not observe the development of plastic defor- mations near the perforation interval in the upper aquifer, while they are localized to the lower left part of the salt layer only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The main fracture opens on asperities along the depth intervals corresponding to the lower aquifer and the top part of the upper aquifer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The CO2 plume shape in the modified model shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 25a slightly differs from that obtained in the base case rightward to the fault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The alteration in the gas saturation distribution is attributed to the open fracture, along which a small portion of carbon dioxide flows upward and then laterally at the top of the upper aquifer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The gas saturation reaches larger values at the lower aquifer on the right side of the fault in the modified model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='2, we describe how the mechanical prop- erties (bulk modulus, shear modulus, cohesion, angle of internal friction, and dilatancy) are set in the fault zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' All parameters except for dilatancy decrease by 30% towards the fault core as compared to the corresponding parameters of the host rock at the considered depth, while the dilatancy grows by the same quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In the current numerical exper- iment for the sensitivity analysis, we carry out the coupled simulations of CO2 injection assuming the alteration of the mechanical properties in the fault zone by 80% towards its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Analyzing the results of simulations we conclude that there are plastic deformations developed in the fault zone along its entire length in addition to the similar domains near the perforation interval in the upper aquifer and in the bottom left part of the salt layer as in the base case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The main fracture opens along the entire length;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' however, its aperture is smaller as compared to the case of the modified coupled model with the reduced values of the mechanical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We find that the improvement of permeability in the fault zone due to the mechanical effects is negligible leading to the same shapes of the CO2 plume and close values of the injected mass of carbon dioxide in the modified and base cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The comparison of the solutions in terms of the injected CO2 mass is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 24 (dashed blue line versus red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Finally, we conduct a coupled modeling of CO2 injection into storage aquifers considering the fault zone with the uniform permeability set to 10 mD so that in the current experiment, we do not distinguish the damage and core zones in terms of the flow properties and present the fault zone as a conduit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The carbon dioxide injected mass obtained using the modified model exceeds that obtained in the base case (see 24, orange line compared to the red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In the modified model, plastic deformations are developed in the same zones as in the base case, namely, near the perforation interval in the upper aquifer and in the bottom part of the salt layer close to the left border of the formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Similar to the base case, the main fracture does not open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The increased permeability in the fault zone contributes to the flow of a larger volume of CO2 in parallel and perpendicular directions to the main crack resulting in the slightly greater size of the carbon dioxide plume rightward to the fault as compared to the base case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Summary and conclusions In this paper, we developed a coupled hydro-geomechanical model for the simulation of CO2 injection (storage) into a saline aquifer intersected by a tectonic fault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The model is based on the reservoir simulator MUFITS, mechanical simulator FLAC3D, and the in-house algorithm performing the two-way coupling of the simulators, namely, pressure, temperature, and density distributions are transferred from MUFITS to FLAC3D, while porosity and permeability fields estimated with on the basis of deformations and stresses are passed from FLAC3D to MUFITS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The latter calculation relies on the novel mathematical model proposed in the current study describing the dynamics of the permeability alteration inside the tectonic fault domain composed of the damage zone and fault core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' During the modeling of the CO2 injection, MUFITS solves a dynamic problem of the non-isothermal multiphase flow in a rock formation, while FLAC3D is applied to solve the quasi-static problem and computes the mechanical equilibrium of the reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We simulated the CO2 storage via the developed coupled approach at the example of two-dimensional synthetic and realistic reservoir models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We verified the proposed coupled model by solving the problem of transient flow of slightly compressible fluid to a vertical well fully penetrating the infinite-acting reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Firstly, we compared the results of numerical simulations with an analytical solution in terms of distributions of pres- sure, stress tensor components, and vertical displacement preserving the true diffusivity in the hydrodynamic simula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Secondly, we accounted for the alteration of porosity Preprint submitted to Journal of Natural Gas Science and Engineering Page 21 of 26 Geomechanical risks of CO2 storage Figure 25: Results of coupled hydro-geomechanical modeling of CO2 injection into the upper and lower aquifers at a fixed bottomhole pressure of 600 bar and decreased (as compared to base case described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1) values of the mechanical properties of the storage aquifers and caprock as descibed in Table 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' plots (a) and (b) show the gas saturation and volumetric strain fields at the end of the simulation period (30 years), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' and permeability in the hydrodynamic model depending on the volumetric strain evaluated by the mechanical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In this numerical experiment, we compared the results of MU- FITS+FLAC3D and FLAC3D in terms of the parameters listed above and a good match is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Using the synthetic reservoir model, we examined two locations of the storage aquifer at a depth of 950 m and 2 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' For each configuration we demonstrated the results of modeling corresponding to the formation with no tectonic stresses and at pronounced tectonic stresses applied in the initial mechanical reservoir state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We observed that in the absence of the tectonic stresses, plastic deformations do not develop in the reservoir, and the major fracture in the fault core opens on asperities in the elastic mode contributing to an increase in the fault zone permeability along its plane and CO2 leakage out of the target aquifer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' When the tectonic stresses are pronounced, we found that the plastic deforma- tions are developed in the fault zone in addition to the major crack opening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' It results in a permeability increase in the directions along and perpendicular to the fault, CO2 leakage into the upper aquifer, and considerably larger mass of the injected carbon dioxide as compared to that obtained in the case with no tectonic stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In the case of the realistic reservoir, we consider a slice of the formation sector and analyzed the CO2 injection with constant bottomhole pressure of 350 bar (base value) and 600 bar (upper limit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We determined the absence of undesirable mechanical effects in the base case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' For the increased value of the bottomhole pressure, we demonstrated that the fault remains stable while plastic deformations are developed in the vicinity of the perforation interval indi- cating the possible initiation of hydraulic fractures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Further, using the realistic reservoir model, we performed the sensi- tivity analysis of the coupled model to the input parameters describing the fault behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We varied the mechanical properties of the storage layers and fault zone as well as the fault permeability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' It was shown that the reduced values of the mechanical properties in the target layers contribute to an increase in the volumetric deformations (leading to an increase in porosity and permeability) and a partial opening of the main fracture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Strengthened contrast in the mechanical parameters of the host rock and fault core yields insignif- icant plastic deformations in the fault zone and the main fracture opening with a negligibly small aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Finally, an increased permeability in the fault zone results in a tangible increase in the injected CO2 mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Acknowledgements The authors are grateful to the management of Gazprom- neft Science & Technology Center for organizational and financial support of this work, in particular to Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Oleg Ushmaev, Nikolay Glavnov, Evgeny Sergeev and Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Mars M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Khasanov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Constitutive relations to a medium with internal friction and dilatancy Prandtl-Rice constitutive relations to a medium with internal friction and dilatancy are formulated in Nikolaevskii (1971) as follows: 푑푒푖푗 = Π푖푗푘푙푑휎푘푙,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (46) where Π푖푗푘푙 = [ − 휈 2퐺(1 + 휈)훿푖푗훿푘푙 + 1 4퐺 (훿푖푘훿푗푙 + 훿푘푗훿푖푙 )] + 1 4퐻 ( 푁푖푗 + 2 3Λ훿푖푗 ) ( 푁푘푙 + 2 3훼훿푘푙 ) (47) 푁푖푗 = 푠푖푗∕푇 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 푇 = (푠푖푗푠푖푗 )1∕2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 푠푖푗 = 휎푖푗 − 훿푖푗휎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 휎 = 1 3휎푖푖 Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 푑푒푖푗 and 푑휎푘푙 are components of strain and stress tensor increments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 퐺 and 휈 are shear modulus and Poisson coefficient, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 푠푖푗 are components of deviatoric Preprint submitted to Journal of Natural Gas Science and Engineering Page 22 of 26 a 1400 b) 1400 1500 1500 1600 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e-01 1600 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='5e-03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='6 1700 1700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='001 1800 1800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='4 1900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='3 1900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0005 三 -2000 = -2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='2 N-2100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='1 N-2100 0 2200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e+00 2200 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='0e-04 2300 2300 2400 2400 2500 2500 2600 2600 2700 2700 0 200 400 600 800 1000 12001400 0 200 400 600 800 1000 1200 1400 x,m x,mGeomechanical risks of CO2 storage stress tensor and 푇 is the shear stress intensity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Λ is dilatancy coefficient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 훼 is internal friction coefficient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' in the tensor ex- pressions formulated above we use the standard convention on summation over repeating indexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The alternative form of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (46) and (47) is formulated in Rudnicki and Rice (1975): Δ휎푖푗 = 퐸푖푗푘푙Δ휀푘푙, (48) 퐸푖푗푘푙 = 퐺 {[(훿푖푘훿푗푙 + 훿푖푙훿푘푗 )+ (퐾 퐺 − 2 3 ) 훿푘푙훿푖푗 ] − − 퐺 (퐻 + 퐺) + 훼Λ퐾 ( 푁푖푗 + 퐾 퐺 Λ훿푖푗 ) (푁푘푙+ 퐾 퐺 훼훿푘푙) } , (49) where 퐾 = 2(1 + 휈)퐺∕[3(1 − 2휈)]is the bulk modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Implementation of the fault zone permeability into the hydrodynamical model In the main text, we introduce three permeability types: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' permeability of the host rock – 푘, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (13), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' permeability of the system of the natural fractures in the damage zone of the fault – 푘푓, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (28), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' permeability of the main fracture opened on asperities – 푘푐, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In the current section, we describe how parameters 푘, 푘푓, and 푘푐 are combined in the hydrodynamical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We begin with the cells related to the damage zone (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 26a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Each cell includes two interpenetrating isotropic Figure 26: The figure shows the representations of the cells belonging to the damage zone of the tectonic fault (panel a) and to its core (panel b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' continua, namely, host rock and natural fractures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' We apply the equation describing the total permeability of the layered formation in the direction parallel to the stratification: ̄푘 = ∑ 푖 푘푖ℎ푖 ∑ 푖 ℎ푖 , (50) where 푘푖 and ℎ푖 are the permeability and the thickness of each layer in the layered reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' As a result, we estimate the permeability of the cells located in the damage zone of the fault as follows: 푘푥 = 푘푧 = 푘(1 − 휀푝) + 푘푓휀푝, (51) where 휀푝 is plastic volumetric strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Next, we move to the cells related to the fault core (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' 26b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Each of them is intersected by a main crack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' In the derivations, we do not account for the fracture inclination assuming that the tectonic fault is approximately vertical and parallel to z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Leftward and rightward to the fracture, the reservoir structure is similar to the damage zone, which is the host rock containing the network of the natural fractures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' The main crack opening impacts on the cell permeability in the vertical direction only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' (50), we derive the expressions for the total permeability along x and z-axis as follows: 푘푥 = 푘(1 − 휀푝) + 푘푓휀푝, 푘푧 = 푘푥(1 − 퓁) + 푘푐퓁, 퓁 = 푤푐∕퐿, (52) where 푤푐 is the geometrical aperture of the main crack, 퐿 is the cell size along x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' References Aagaard, B.' metadata={'source': 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system’s openness under conditions of changes in the crack roughness coefficient based on data on the stress-strain state [in russian].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Drilling and oil , 30–38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Zhou, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', Birkholzer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', Tsang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} 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+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', Gorelick, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=', 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Earthquake triggering and large- scale geologic storage of carbon dioxide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences 109, 10164–10168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} +page_content=' Preprint submitted to Journal of Natural Gas Science and Engineering Page 26 of 26' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE4T4oBgHgl3EQfKgzs/content/2301.04931v1.pdf'} diff --git a/RtAzT4oBgHgl3EQfXPx9/content/tmp_files/2301.01315v1.pdf.txt b/RtAzT4oBgHgl3EQfXPx9/content/tmp_files/2301.01315v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..aad8496233cc1102b8fccc0166143f86d6ed962e --- /dev/null +++ b/RtAzT4oBgHgl3EQfXPx9/content/tmp_files/2301.01315v1.pdf.txt @@ -0,0 +1,1326 @@ +Neural SDEs for Conditional Time Series Generation +and the Signature-Wasserstein-1 metric +Pere Díaz Lozano +Departament de Matemàtiques +Universitat Autònoma de Barcelona +pere.diaz@uab.cat +Toni Lozano Bagén +Departament de Matemàtiques +Universitat Autònoma de Barcelona +tonilb@mat.uab.cat +Josep Vives +Departament de Matemàtiques i Informàtica +Universitat de Barcelona +josep.vives@ub.edu +January 5, 2023 +Abstract +(Conditional) Generative Adversarial Networks (GANs) have found great success +in recent years, due to their ability to approximate (conditional) distributions over +extremely high dimensional spaces. However, they are highly unstable and computa- +tionally expensive to train, especially in the time series setting. Recently, it has been +proposed the use of a key object in rough path theory, called the signature of a path, +which is able to convert the min-max formulation given by the (conditional) GAN +framework into a classical minimization problem. However, this method is extremely +expensive in terms of memory cost, sometimes even becoming prohibitive. To overcome +this, we propose the use of Conditional Neural Stochastic Differential Equations, which +have a constant memory cost as a function of depth, being more memory efficient than +traditional deep learning architectures. We empirically test that this proposed model +is more efficient than other classical approaches, both in terms of memory cost and +computational time, and that it usually outperforms them in terms of performance. +Keywords: conditional generative modelling, neural networks, expected signature, rough +path theory, Wasserstein generative adversarial networks, neural stochastic differential equa- +tions +arXiv:2301.01315v1 [stat.ML] 3 Jan 2023 + +1 +Introduction +The simplest approach to perform time series forecasting is to only be interested in a determin- +istic response, which is usually thought of as the mean of possible outcomes. Consequently, +the model is unable to report any inherent uncertainty, which is a major shortcoming for +most real world applications. +Recent work has ought to solve this by turning their attention to Generative Adversarial +Networks (GANs) (Arjovsky et al., 2017, Goodfellow et al., 2014). Their basic idea is to +train two networks against each other: +• The generator: it generates samples from a distribution by sending random noise +through a parameterized model. +• The adversarial network: its job is to approximate a loss function between the real +data distribution and the distribution produced by the generator. +The training algorithm consists in training the adversarial network for a certain numbers +of steps, and once a good enough approximation of the loss function we are interested in is +reached, then perform one step on the parameters of the generator. +The original GAN formulation was set such that the adversarial network approximated the +Jensen Shannon divergence (Goodfellow et al., 2014). However, in our case we will focus on +the Wasserstein GAN formulation, where the adversarial network (called the critic) aims at +approximating the Wasserstein-1 distance (Arjovsky et al., 2017). +Although GANs have had great success in approximating probability measures over extremely +high dimensional spaces, especially in Computer Vision, they are very unstable to train. +Moreover, the fact that one needs to train an adversarial network before performing one step +on the generator means that the training time is considerably large. +1.1 +Signature-Wasserstein-1 metric +A more efficient approach would be to approximate the loss function in a closed form way, +without the need of an iterative method. In this direction, Ni et al., 2021 proposed the use of +the signature transform, a fundamental object from rough path theory which captures many +of the most important analytic and geometric properties of a given path1. By using it, one is +able to derive an expression that is able to approximate uniformly well the Wasserstein-1 +distance between two distributions on path space in a closed form non-parametric way. In +doing so, all of the drawbacks that are inherent to the (Wasserstein) GAN formulation are +overcome, while still maintaining its strong theoretical guarantees. This new framework is +called the Signature-Wasserstein GAN (SigWGAN). +1Although in practice we are usually given a set of discrete streams of data, we can always embed them +into the set of continuous paths by performing some kind of interpolation scheme. Therefore, we will not +really differentiate between these two concepts. +1 + +In Ni et al., 2020, the authors proposed the Conditional Signature-Wasserstein GAN (SigCW- +GAN), which is a modification of the SigWGAN to learn instead conditional distributions, as +the ones we are interested in when doing non-deterministic time series forecasting. However, +since this algorithm needs to perform a Montecarlo procedure for each data sample, the +memory cost is dramatically increased, sometimes even becoming prohibitive. They elude this +by assuming that the distribution of the observed time series has an autoregressive structure, +with the next observed value depending only on the previous q > 0, with q relatively small. +1.2 +Contributions +In order to overcome this increase in the memory cost, we propose the use of Neural Stochastic +Differential Equations (Neural SDEs) (see Kidger, Foster, Li, Oberhauser, et al., 2021, Li +et al., 2020), which are essentially generative models formed by classical stochastic differential +equations whose vector fields are parameterized by neural networks. The main advantage +over traditional deep learning architectures is the fact that one is able to backpropagate +through a Neural SDE without the need to store the intermediate quantities produced in the +forward pass, being more memory efficient than traditional deep learning models. +By encoding the path we want to condition on, and by making the initial condition of the +Neural SDE depend on this codification, we introduce what we call Conditional Neural +Stochastic Differential Equations (CNSDEs), which produce conditional distributions in path +space. +When using all of the above, we are able to formulate a model and a training algorithm +for approximating conditional distributions on path space, which is both mathematically +well founded and practically more efficient. Furthermore, we will test it against other more +traditional baselines, concluding that in most cases it outperforms them according to several +metrics. +2 +Background +2.1 +The Signature-Wasserstein-1 metric +Let sig(x), sigN(x) denote the signature and the truncated signature of order N of a +continuous path x, respectively, both with the basepoint and time augmentations (see Morrill +et al., 2020). +Let µ, ν be two distributions in path space. The Kantorovich-Rubinstein duality states that +the Wasserstein-1 distance can be calculated as +W1(µ, ν) = +sup +∥F∥L≤1 +� +Ex∼µ[F(x)] − Ex∼ν[F(x)] +� +where the supremum is over all the 1−Lipschitz functions. +2 + +In Ni et al., 2021, Ni et al., 2020 the authors use the signature transform and its universal +non-linearity to derive a closed form expression that uniformly approximates the Wasserstein-1 +distance, +sigNW1(µ, ν) = +���Ex∼µ[sigN(x)] − Ex∼ν[sigN(x)] +��� +2 +(1) +where ∥·∥2 denotes the L2 norm. This approximation is called the truncated Signature- +Wasserstein-1 metric of order N. +Sections A and B in the supplementary material provide an in depth derivation of (1). +2.2 +The Conditional Signature-Wasserstein GAN algorithm +In the unconditional case, whenever we intend to compute (1) from a set of given data, we +can simply approximate both expected signatures by their empirical average, by performing +a Montecarlo simulation. +However, if instead we are interested in approximating conditional distributions, then one is +not able to perform a Montecarlo simulation to approximate the expected signatures of the +conditional distributions given by the data, since most of the times we are only handled one +sample. +By using many of the properties of the signature, one is able to prove that the conditional +expected truncated signature can be approximated arbitrarily well by applying a linear map +on the truncated signature of the conditioning path x, +ˆEy∼Y |x[sigM(y)] = ˆℓ(sigN(x)) +(2) +which can be approximated from the data by standard linear regression techniques. An in +depth derivation of (2) is given in Section C. +The resulting training algorithm is called the Conditional Signature-Wasserstein GAN (SigCW- +GAN) (see Ni et al., 2020, Algorithm 2). +Figure 1: Flowchart of the SigCWGAN algorithm. +3 + +Input path a +Ey~Yla[sigM(y)] +E. +e(sigN (α)) +SigN-Wi +metric +Z noise +Conditional Generator +Ey~Go(Z,a)[sig M (y)] +distribution +Ge(z,α) +Monte Carlo2.2.1 +Disadvantages of the SigCWGAN algorithm +One of the main drawbacks of the SigCWGAN algorithm is the Montecarlo procedure needed +to estimate the expected signature of the conditional generator for each sample in the +minibatch, which considerably increases the memory cost. This is in contrast to the general +conditional GAN setting, where both the generator and the critic simply take as input the +corresponding sample x (Mirza & Osindero, 2014). Therefore, the conditional Wasserstein +GAN algorithm (CWGAN) has the same memory cost as the unconditional WGAN approach. +This means that the SigCWGAN algorithm consumes much more memory that the traditional +CWGAN approach, and as a consequence sometimes we need to decrease the batch size or +the complexity of the model. As we will see in the following pages, this can be remedied by +using a Neural Stochastic Differential Equation as a generator, which are much more memory +efficient to train than traditional neural networks architectures. +2.3 +Neural Stochastic Differential Equations +Neural Stochastic Differential Equations (Neural SDE) are models that arise from the union +of stochastic differential equations and neural networks, two of the biggest paradigms in +mathematical modelling, resulting in generative models that produce distributions in path +space. +The basic structure of a Neural SDE is +Z0 ∼ ξθ(V ) +dZt = fθ(t, Zt)dt + gθ(t, Zt) ◦ dWt +Yt = αθZt + βθ +with V ∼ N(0, Idv) drawn from a dv−dimensional standard normal distribution, ξθ, fθ, gθ +being neural networks and αθ, βθ being matrices of learnable weights. Wt denotes the Wiener +process, and the ◦ indicates that the integration is in the Stratonovich sense. +Z represents the hidden state of the model. If it was the output, then the resulting stochastic +process would satisfy a Markov property, which does not need to be true in general. This is +the reason for the final readout linearity. +As we already said, the solution to a stochastic differential equation is a distribution in path +space. However, just like with ODEs, computing it in an analytical form is almost always +impossible. Nonetheless, one can sample from it. This is what a numerical SDE solver does, +returning a sampled path evaluated at a set of discrete time locations. +Thus a Neural SDE fits the GAN setting, since evaluating its density is not possible and it is +able to generate samples by sending random noise (in the form of the initial condition and +the Wiener process) through a parameterized model. +4 + +2.3.1 +Backpropagation through a Neural SDE +If we intend to fit a Neural SDE to some data, we need an algorithm to compute gradients +with respect to its parameters. We will briefly summarize two ways to do this: +Discretize-then-optimize: The most straightforward way consists in performing the usual +backpropagation, differentiating through the internal operations of the differential equation +solver, and therefore needing to store them in the forward pass. In the literature this is +commonly referred to as discretize-then-optimize, because we are directly optimizing the +discretization we are using in practice. +If we denote by H the memory cost of recording the operations of one solver step, and by +N the number of steps, then the discretize-then-optimize method consumes about O(HN) +memory. Notice that this is the memory cost of traditional deep learning models, such as +Recurrent Neural Networks. +Reversible solvers: Alternatively, if we use a reversible solver, we dramatically reduce the +memory cost. Consider a differential equation solver, which in the forward pass iteratively +computes the next step from the previous one, +(zti, αti) �→ (zti+1, αti+1), +(3) +where {zti}n +i=1 is the numerical approximation of the solution to some differential equation, +and αti represents the intermediate quantities that the solver needs to store at step i in order +to compute the solution at the next step i + 1. +Then, a solver is said to be algebraically reversible if it can compute the previous step from +the next step, +(zti+1, αti+1) �→ (zti, αti), +(4) +by using a closed-form expression. Notice how, in this case, the intermediate values (zti, αti) +do not need to be stored in memory, and they can be recomputed in the backward pass. In +this case, the memory cost is only O(H). +The full algorithm, along with the only known such SDE solver (called the reversible Heun +method), can be found in Kidger, Foster, Li, and Lyons, 2021. +3 +Conditional Neural Stochastic Differential Equations +In our case we are interested in generating distributions that are conditioned on some input +path x. We propose to condition a Neural SDE by making the initial condition of the SDE +depend on the path we want to condition on, following an encoder-decoder structure. +Definition 1 (Conditional Neural Stochastic Differential Equation). Let +ξθ : Rdh → Rdz +fθ : [0, T] × Rdz → Rdz +gθ : [0, T] × Rdz → Rdz×dw +5 + +be feedforward neural networks. Let φ : T S([0, τ]; Rdx) → Rdh be a continuous map (with +or without learnable parameters), with T S([0, τ]; Rdx) being the space of time series with +timestamps in [0, τ] and dx−dimensional observations. Then we define a Conditional Neural +Stochastic Differential Equation (CNSDE) as +h = φ(x) +Z0 = ξθ(h) +dZt = fθ(t, Zt)dt + gθ(t, Zt) ◦ dWt +Yt = αθZt + βθ +where αθ ∈ Rdy×dz and βθ ∈ Rdy are matrices of learnable weights. +The following is an overview of a CNSDE. An input path x (red) is fed into the model, which +determines the initial condition of the SDE z0. Then a trajectory w is sampled from the +Wiener process (blue). All of this is fed to the differential equation solver, which gives us the +solution z (green). After that we apply a final readout linearity, which produces the output +of the model y (purple). As a summary, y is a sample from a distribution in path space that +is conditioned on a given sample x. Whenever we change x, this distribution changes. +Figure 2: Overview of a Conditional Neural SDE. +Notice how the initial condition of the SDE is completely determined (once the parameters +are fixed) by the input path x. An alternative approach is to add a random component to +the initial condition Z0, which usually enforces diversity and improves learning. This can be +simply done by setting the initial condition to be +h = φ(x) +(5) +Z0 ∼ [ξ1 +θ(h), ξ2 +θ(V )] +(6) +with ξ1 +θ : Rdh → Rdz−k and ξ2 +θ : Rdv → Rk being neural networks and V ∼ N(0, Idv). The +larger the size of k with respect to dz, the more we will be enforcing diversity. +6 + +M~m +y=αe+4 +Empirical Analysis +The goal of this section is to compare the performance of the SigCWGAN algorithm and the +CNSDE against some more traditional approaches which will serve as a baseline. All the +code is available in the GitHub repository https://github.com/pere98diaz/Neural-SDEs-for- +Conditional-Time-Series-Generation-and-the-Signature-Wasserstein-1-metric. +We will compare the performance of three models: +LSTM Conditional Wasserstein GAN: the model that will serve as a pure baseline is +based on Long short-term memory (LSTM) networks (Hochreiter & Schmidhuber, 1997). It +is trained by using the Conditional Wasserstein GAN algorithm, with the critic being also +formed by LSTMs. +LSTM Conditional Signature-Wasserstein GAN: the second model is formed by the +same conditional generator as the LSTM CWGAN model, but using the truncated Signature- +Wasserstein-1 metric to approximate the Wasserstein-1 distance. To train the model we use +the SigCWGAN algorithm. +Neural SDE Conditional Signature-Wasserstein GAN: the third model is formed by a +Conditional Neural Stochastic Differential Equation as the generator, and uses the truncated +Signature-Wasserstein-1 metric to approximate the Wasserstein-1 distance. We also use the +SigCWGAN algorithm for training. +4.1 +Resources cost +In this section we will be comparing the three models defined earlier in terms of two key +resources: maximum memory allocated on the GPU (left hand side) and computational time +required to perform one step on the generator (right hand side). The architectures of the +three models are chosen so that they have a similar number of parameters. +Figure 3: Memory consumption (MB) and time per step (s) for the three models. +We can see how, in terms of memory cost, the LSTM trained with the SigCWGAN algorithm +is by far the most expensive one. This is mainly because of the Montecarlo procedure we +need to perform for each sample on the minibatch. However, notice that when using the +7 + +Memory cost +Time per step +17.5 +DO +15.0 +12.5 +step +DO+ +10.0 +per! +7.5 +4000 +5.D +2400 +25 +0 +0.D +LSTM CWGAN +LSTM SigCWGAN CNSDE SigCWGAN +LSTM CWGAN +LSTM SigCWGAN CNSDE SigCWGANCNSDE as the conditional generator, the memory drops considerably, due to the use of a +reversible solver to perform backpropagation. Unsurprisingly, the LSTM trained with the +CWGAN algorithm is the cheapest one. +In terms of time per generator step, the most expensive model is the LSTM trained with the +CWGAN algorithm. This is mainly because of the fact that we need to perform k steps on +the parameters of the critic before performing one step on the parameters of the generator (in +our case, we picked the standard default value, which is k = 10). The models trained with the +SigCWGAN algorithm approximate the Wasserstein-1 loss in a closed form non-parametric +way, which is the reason why it takes much less time to perform one generator step. +In conclusion, the most balanced model out of the three is the CNSDE trained with the +SigCWGAN algorithm. +4.2 +Experiments +We performed experiments on four one dimensional different datasets, which will be described +next. Each of the three models were trained three times, and we measured their performance +in terms of the following statistics, all evaluated in an out-of-time test set. +Classification error: We train an LSTM whose task is to, given a pair of input/output +streams, classify whether the output stream is real or generated. The architecture of this +classifier is the same for each model. The worse the performance of this classifier, the better +the generative model is. +Higher order Signature-Wasserstein-1 metric: Similarly to the SigCWGAN algorithm, +we compute the L2-distance between the predicted and generated expected signature. However, +in this case we truncate the signatures at higher levels than the ones we used during training: +depth 6 for the input paths and depth 5 for the output paths. +Unordered Wasserstein-1 metric: We compare the Wasserstein-1 distance between the +real one dimensional distributions that are given by: a) taking the data points yt from the +output streams, without considering that they are ordered in time. b) taking the differences +between the data points in the output streams and the last value from the input stream, +yt − xT, also without considering that they are ordered in time. c) For each stream, taking +the largest difference given by the procedure in b). d) For each stream, taking the smallest +difference given by the procedure in b). +Extreme values metric: Consider the empirical distribution given by the procedure we +detailed in the Unordered Wasserstein-1 metric c). Let q indicate the value of a very high +percentile of it, for example 95%. Then we estimate the probability, conditioned on the input +path x, of producing a path y such that maxt yt − xT is equal or over q. In some cases we +will instead consider percentage increases, as maxt(yt − xT)/xT. +Just like we defined a way of measuring how good the model can detect a high possibility +of an extremely large increment, we can do the same for an extremely value decrease, by +8 + +considering mint yt − xT instead (or mint(yt − xT)/xT) and setting the percentile to be very +low, like 5%. +We should remark that evaluating the performance of a conditional generative model, especially +in the time series framework, is very challenging, and none of the above statistics should be +considered as ground truth metrics. Moreover, most of the time the metric we are interested +in depends on the problem we have at hand, and the purpose and use we want to give to +that model. +4.2.1 +AR(5) data +The first dataset that we used was simulated from an autoregressive model of order p = 5. We +considered the problem of predicting the next 40 steps given the previous 60. The training +time was set to a maximum of 2 hours. However, for the models using the SigCWGAN +algorithm one is able to define an early stopping criteria, which was set to 1,000 steps without +improvement. Moreover, at the end of training one can just keep the parameters that gave +the best loss on a validation set. +The next table indicates some general information on training, like the maximum memory +allocated, the training time or the number of steps that were performed on the generator. +The last column indicates the objective function that was used for training in the SigCWGAN +models evaluated in the test set. For completeness, we also show it for the LSTM CWGAN, +although it was not set to explicitly optimize it at all. +Memory (MB) +Time (h) +Steps +Sig-W1 loss +LSTM CWGAN +1945 +2.000 ± 0.000 +566 ± 15 +5.578 ± 2.297 +LSTM SigCWGAN +9931 +1.712 ± 0.499 +8789 ± 949 +2.804 ± 0.048 +NSDE SigCWGAN +3764 +1.555 ± 0.386 +16839 ± 1673 +1.855 ± 0.095 +Table 1: Some general information on the training process, AR(5) dataset. +Next we show the Area Under the Curve (AUC) and the accuracy obtained from training +an LSTM to tell apart real from fake data. We can clearly see how the LSTM CWGAN +outperforms the rest. This is not a surprise at all, since in the GAN framework the generator +is directly competing against another network whose job is precisely to distinguish real data +from generated data. +The best model according to each metric is always marked as bold. +AUC +Accuracy +LSTM CWGAN +0.747 ± 0.144 +0.685 ± 0.121 +LSTM SigCWGAN +0.866 ± 0.053 +0.775 ± 0.043 +NSDE SigCWGAN +0.959 ± 0.021 +0.873 ± 0.031 +Table 2: Classification error, AR(5) dataset. +9 + +HO Sig-W1 metric +EV+ AUC +EV− AUC +LSTM CWGAN +5.412 ± 0.361 +0.831 ± 0.098 +0.834 ± 0.130 +LSTM SigCWGAN +5.092 ± 0.005 +0.958 ± 0.039 +0.960 ± 0.035 +NSDE SigCWGAN +1.074 ± 0.083 +0.988 ± 0.004 +0.982 ± 0.013 +Table 3: The Higher order Signature-Wasserstein-1 metric and both the Extreme values +metrics, AR(5) dataset. The selected percentiles for the EV+ and EV− were 99% and 1%, +respectively. +W1-a +W1-b +W1-c +W1-d +LSTM CWGAN +0.092 ± 0.127 +0.061 ± 0.075 +0.216 ± 0.296 +0.236 ± 0.301 +LSTM SigCWGAN +0.086 ± 0.016 +0.056 ± 0.004 +0.155 ± 0.089 +0.135 ± 0.059 +NSDE SigCWGAN +0.036 ± 0.015 +0.017 ± 0.005 +0.156 ± 0.035 +0.165 ± 0.061 +Table 4: The four unordered Wasserstein-1 metrics, AR(5) dataset. +In conclusion, we see in Table 1 that the Neural SDE did a better job minimizing the Sig-W1 +loss function than both LSTMs. This did not translate into a better performance in terms +of the Classification error in Table 2, meaning that probably the Signature-Wasserstein-1 +metric failed to produce a good enough approximation of the codification of the conditioned +stochastic process. However, in terms of the rest of the metrics, the Neural SDE outperforms +the other models, implying that it was able to encode many of its most important geometric +properties. +4.2.2 +Seattle weather +The second dataset was formed by daily observations of the maximum temperature reached +in Seattle from 1948 to 20172. Our task will be to, given the last 60 days, predict the next 30. +The training time was set to a maximum of 3 hours. For the models using the SigCWGAN +we set an early stopping criteria of 1,000 steps without improvement. +Memory (MB) +Time (h) +Steps +Sig-W1 loss +LSTM CWGAN +2055 +3.000 ± 0.000 +965 ± 6 +6.944 ± 0.360 +LSTM SigCWGAN +8881 +0.937 ± 0.201 +7501 ± 1639 +3.119 ± 1.009 +NSDE SigCWGAN +4696 +1.317 ± 0.460 +4834 ± 1626 +1.410 ± 0.067 +Table 5: Some general information on the training procedure, Seattle Weather dataset. +2The dataset can be found in https://www.kaggle.com/datasets/rtatman/did-it-rain-in-seattle-19482017 +10 + +AUC +Accuracy +LSTM CWGAN +0.617 ± 0.060 +0.573 ± 0.033 +LSTM SigCWGAN +0.715 ± 0.114 +0.652 ± 0.087 +NSDE SigCWGAN +0.732 ± 0.038 +0.660 ± 0.031 +Table 6: Classification error, Seattle Weather dataset. +HO Sig-W1 +EV+ AUC +EV− AUC +LSTM CWGAN +6.014 ± 0.026 +0.514 ± 0.025 +0.577 ± 0.047 +LSTM SigCWGAN +5.465 ± 0.265 +0.808 ± 0.099 +0.888 ± 0.046 +NSDE SigCWGAN +1.032 ± 0.082 +0.905 ± 0.001 +0.940 ± 0.003 +Table 7: The Signature-Wasserstein-1 metric and both the Extreme values metrics, Seat- +tle Weather dataset. The selected percentiles for the EV+ and EV− were 95% and 5%, +respectively. +W1-a +W1-b +W1-c +W1-d +LSTM CWGAN +0.061 ± 0.007 +0.155 ± 0.018 +0.251 ± 0.018 +0.175 ± 0.018 +LSTM SigCWGAN +0.060 ± 0.011 +0.035 ± 0.006 +0.094 ± 0.030 +0.144 ± 0.072 +NSDE SigCWGAN +0.047 ± 0.002 +0.028 ± 0.001 +0.062 ± 0.012 +0.055 ± 0.008 +Table 8: The four unordered Wasserstein-1 metrics, Seattle Weather dataset. +The conclusions are pretty much similar to the ones we obtained for the AR(5) dataset. In +Table 5 we see that the Neural SDE did a way better job at minimizing the CSig-W loss +function than the LSTMs. Since this did not translate to the Classification error performance +showed in Table 6, this means that the Signature-Wasserstein-1 metric did not do a great job +at codifying the conditional stochastic process. However, since the Neural SDE outperforms +the other models in terms of the rest of the metrics, we also conclude that it succeeded in +capturing many important geometric properties. +We especially highlight the results showed in Table 7, where we can see how the LSTM +trained with the WGAN algorithm completely failed to perform well in terms of detecting +extreme values. This is in contrast to the models trained with the SigCWGAN method, +which did a very good job. +4.2.3 +Forex +The third dataset was a Forex time series formed by observations of the bid price between +the Euro and the Dollar. More specifically, at each timestamp it indicates the highest price a +buyer will pay, in Dollars, to buy one Euro. +The observations are spanned over weeks 19 and 20 of 2020, and are irregularly spaced. The +NSDE SigCWGAN model is able to work with irregularly sampled data, since both the +conditioner and the loss function are based on the signature transform. However, this is not +11 + +the case for the other two models. In order to be able to compare them, we aggregated the +data by computing the mean in each 30 seconds interval. +Our task will be to, given the last 80 observations, predict the next 80. The training time +was set to a maximum of 4 hours. For the models using the SigCWGAN algorithm, we set +an early stopping criteria of 1,000 steps without improvement. +Memory (MB) +Time (h) +Steps +Sig-W1 loss +LSTM CWGAN +5818 +4.000 ± 0.000 +718 ± 6 +4.860 ± 1.947 +LSTM SigCWGAN +9209 +1.555 ± 0.252 +10417 ± 1664 +2.491 ± 0.002 +NSDE SigCWGAN +8697 +1.653 ± 0.236 +3001 ± 433 +2.049 ± 0.019 +Table 9: Some general information on the training procedure, Forex dataset. +AUC +Accuracy +LSTM CWGAN +0.962 ± 0.087 +0.932 ± 0.097 +LSTM SigCWGAN +0.838 ± 0.104 +0.768 ± 0.118 +NSDE SigCWGAN +0.630 ± 0.093 +0.587 ± 0.060 +Table 10: Classification error, Forex dataset. +HO Sig-W1 +EV+ AUC +EV− AUC +LSTM CWGAN +4.614 ± 1.346 +0.500 ± 0.004 +0.510 ± 0.017 +LSTM SigCWGAN +3.010 ± 0.004 +0.488 ± 0.011 +0.499 ± 0.005 +NSDE SigCWGAN +2.704 ± 0.077 +0.596 ± 0.024 +0.544 ± 0.013 +Table 11: The Signature-Wasserstein-1 metric and both the Extreme values metrics, Forex +dataset. The selected percentiles for the EV+ and EV− were 90% and 10%, respectively. +W1-a +W1-b +W1-c +W1-d +LSTM CWGAN +0.103 ± 0.048 +0.107 ± 0.046 +0.053 ± 0.014 +0.146 ± 0.052 +LSTM SigCWGAN +0.012 ± 0.001 +0.020 ± 0.003 +0.029 ± 0.003 +0.037 ± 0.007 +NSDE SigCWGAN +0.013 ± 0.000 +0.019 ± 0.002 +0.028 ± 0.003 +0.015 ± 0.002 +Table 12: The four unordered Wasserstein-1 metrics, Forex dataset. +Notice how, in contrast to the last two experiments, the NSDE SigCWGAN model clearly +outperformed the rest of the models in terms of Classification error. We theorize that this is +due to the increment, in terms of length, of both the input and output streams. However, +further experiments should be conducted to test whether this is indeed the true reason or not. +In this problem we perhaps could be especially interested in knowing whether, given a known +path x, there will be a large increment or reduction in a fixed time period. This is tested +with the Extreme values metric, and the results of each model are shown in Table 11. We +can see how the NSDE based model also outperforms the rest in terms of these metrics. +12 + +4.2.4 +IBEX35 +The final dataset was formed by the daily return (between 1993 and 2022) of the IBEX 35 +(IBerian IndEX), which is the benchmark stock market index of the Bolsa de Madrid, Spain’s +principal stock exchange. +Our task will be to, given the last 30 observations, predict the next 15. The training time +was set to a maximum of 2 hours. For the models using the SigCWGAN algorithm, we set +an early stopping criteria of 1,000 steps without improvement. +Memory (MB) +Time (h) +Steps +Sig-W1 loss +LSTM CWGAN +1786 +2 ± 0.000 +1283 ± 3 +2.837 ± 0.297 +LSTM SigCWGAN +9201 +1.442 ± 0.295 +1334 ± 2919 +1.788 ± 0.020 +NSDE SigCWGAN +5825 +1.647 ± 0.423 +6917 ± 1773 +1.811 ± 0.016 +Table 13: Some general information on the training procedure, IBEX35 dataset. +AUC +Accuracy +LSTM CWGAN +0.844 ± 0.115 +0.765 ± 0.104 +LSTM SigCWGAN +0.661 ± 0.066 +0.618 ± 0.052 +NSDE SigCWGAN +0.588 ± 0.074 +0.562 ± 0.058 +Table 14: Classification error, IBEX35 dataset. +HO Sig-W1 metric +EV+ AUC +EV− AUC +LSTM CWGAN +4.910 ± 0.231 +0.776 ± 0.041 +0.507 ± 0.118 +LSTM SigCWGAN +4.606 ± 0.043 +0.867 ± 0.042 +0.642 ± 0.048 +NSDE SigCWGAN +1.301 ± 0.367 +0.886 ± 0.005 +0.687 ± 0.012 +Table 15: The Signature-Wasserstein-1 metric and both the Extreme values metrics, IBEX35 +dataset. The selected percentiles for the EV+ and EV− were 95% and 5%, respectively. +It is interesting to mention that, in terms of the Extreme values metric, the performance of +the models considerably drops during the COVID-19 years. For example, for the Neural SDE +the results of the EV− AUC in the period 2015-2019 is 0.758 ± 0.033, while in the period +2020-2022 is 0.633 ± 0.004. +W1-a +W1-b +W1-c +W1-d +LSTM CWGAN +0.025 ± 0.009 +0.021 ± 0.004 +0.041 ± 0.019 +0.023 ± 0.013 +LSTM SigCWGAN +0.014 ± 0.002 +0.022 ± 0.001 +0.027 ± 0.004 +0.025 ± 0.001 +NSDE SigCWGAN +0.014 ± 0.001 +0.017 ± 0.001 +0.029 ± 0.001 +0.016 ± 0.002 +Table 16: The four unordered Wasserstein-1 metrics, IBEX35 dataset. +We can conclude that in this case the NSDE SigCWGAN clearly outperformed the rest of +the models in terms of all metrics. +13 + +5 +Conclusion +In this paper, we have proposed the use of a Conditional Neural Stochastic Differential +Equation as a conditional generator in the SigCWGAN algorithm, which offsets the great +increase in terms of memory cost produced by the Montecarlo procedure needed for every +sample in the minibatch. +We then tested in practice what was the gain and loss, in terms of computational time +and memory cost, of first replacing the traditional WGAN algorithm with the SigCWGAN +method, and then the traditional LSTM generator with a Neural SDE. We clearly showed +that Neural SDEs trained with the SigCWGAN algorithm were the most balanced in terms +of both resources cost. Finally, we compared their performance in four experiments with four +different datasets, which allowed us to see that, in most cases, the Neural SDEs captured +better some of the properties of the real datasets. +Acknowledgements +The research of Josep Vives is partially supported by Spanish grant PID2020-118339GB-100 +(2021-2024). +Declarations of Interest +All authors report no conflicts of interest. The authors alone are responsible for the content +and writing of the paper. +References +Arjovsky, M., Chintala, S., & Bottou, L. (2017). Wasserstein generative adversarial networks. +In D. Precup & Y. W. Teh (Eds.), Proceedings of the 34th international conference on +machine learning (pp. 214–223). PMLR. +Arribas, I. P. (2018). Derivatives pricing using signature payoffs. arXiv e-prints, arXiv:1809.09466. +Chevyrev, I., & Kormilitzin, A. (2016). 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Wallach, H. Larochelle, +A. Beygelzimer, F. d’Alché-Buc, E. Fox, & R. Garnett (Eds.), Advances in neural +information processing systems 32 (pp. 8024–8035). Curran Associates, Inc. +15 + +Villani, C. (2009). Optimal transport: Old and new. Grundlehren der mathematischen Wis- +senschaften. Springer, Berlin. +16 + +Supplementary Material +A +The signature transform +For completeness, we will give an elementary introduction to the signature transform, and +we will briefly see many of the properties that make it such a convenient transformation in +machine learning. +A.1 +Paths of bounded variation +Definition 2 (p−variation). Let p ≥ 1 be a real number, and ∥·∥ any norm on Rd. Let +x : [0, T] → Rd be a continuous path. The p−variation of x is defined as +∥x∥p = +� +� sup +D +n−1 +� +i=0 +||xti+1 − xti||p +� +� +1/p +(7) +where the supremum is taken over the set of partitions of [0, T], denoted as D. The space of +d−dimensional continuous paths of finite p−variation will be denoted as Vp([0, T]; Rd). +We will equip Vp([0, T]; Rd) with the following norm. +Definition 3 (p−variation norm). The p−variation norm of a path x ∈ Vp([0, T]; Rd) is +defined as +∥x∥p−var = ∥x∥p + sup +t∈[0,T] +∥xt∥ +(8) +where ∥·∥ is any norm in Rd. +For simplicity, we will only work with paths of finite 1−variation, which are called of bounded +variation. The reason for this is that in practice we will always be given a discrete stream +of data, which can be converted into a continuous path by performing some interpolation +scheme. The most common one is linear interpolation (with the resulting paths being of +bounded variation), since then the signature is very easy to compute. This is what the +Signatory package (Kidger & Lyons, 2021) does, providing differentiable computations of the +signature on the GPU. +Moreover, to define a signature of a general path of finite p−variation, we would need to +introduce many new concepts of rough path theory, which is beyond the scope of this paper. +A.2 +The tensor algebra +Definition 4 (Tensor algebra of Rd). Let (Rd)⊗k denote the kth tensor power of Rd. By +convention, (Rd)⊗0 = R. We define the extended tensor algebra of Rd as +T((Rd)) = {a = (a0, a1, ...) | ∀k ≥ 0, ak ∈ (Rd)⊗k} +17 + +We also define the truncated tensor algebra of order N of Rd, which is a linear subspace of +T((Rd)), as +T (N)(Rd) = {a = (a0, a1, ..., aN) | ∀N ≥ k ≥ 0, ak ∈ (Rd)⊗k} +Note that T (N)(Rd) is a real vector space of dimension +N +� +k=0 +dk = +� +� +� +N + 1 +if d = 1 +dN+1−1 +d−1 +if d > 1 +(9) +We will equip T((Rd)) with an admissible norm, defined as follows. +Definition 5. We say that the extended tensor algebra T((Rd)) is endowed with an admissible +norm ∥·∥ if the following conditions hold: +1. For each k ≥ 1, the symmetric group Sk acts by isometry on (Rd)⊗k, i.e. +∥σv∥ = ∥v∥ , +∀v ∈ (Rd)⊗k, ∀σ ∈ Sk +(10) +2. The tensor product has norm 1, i.e. ∀n, m ≥ 1, +∥v ⊗ w∥ ≤ ∥v∥ ∥w∥ , +∀v ∈ (Rd)⊗n, w ∈ (Rd)⊗m +(11) +A.3 +The signature of a path +The signature transform provides a way of encoding any continuous path into an infinite +sequence of statistics, which has many properties that make it a very desirable transformation +in machine learning (Chevyrev & Kormilitzin, 2016). +Definition 6 (Signature of a path). Let x = (x1, ..., xd) : [0, T] → Rd be a continuous path of +bounded variation. The signature of x is defined as the infinite collection of iterated integrals +sig(x) = +� +� +� +· · · +� +0 0, there exists an N ∈ N such that +���sig(x) − sigN(x) +��� ≤ ϵ +(13) +Moreover, if we restrict ourselves to a compact subset K ⊂ V1([0, T]; Rd), then the convergence +is uniformly. +Another very important property of the signature is the universal non-linearity property, +which states that the signature provides a basis for the space of continuous functions on path +space. +Theorem 3 (Universal non-linearity, Arribas, 2018, Theorem 4.2). Let K ⊂ A1([0, T]; Rd) +be a compact subset. Then +� +N∈N +� +x �→ ℓ(sigN(x)) | ℓ : T (N)(Rd+1) → Rv is linear +� +(14) +is uniformly dense in the space of continuous maps from K to Rv, denoted by C(K; Rv). +19 + +A.4 +The signature of a stochastic process +The last property that we will see states that the expected signature of a stochastic process +that has a compact support completely determines its distribution. +Theorem 4 (Fawcett, 2003). Let {Xt}t∈[0,T] and {Yt}t∈[0,T] be two stochastic processes with +compact support K ⊂ A1([0, T]; Rd). Then, we have that +1. The expected signatures are well defined, Ex∼X[sig(x)], Ey∼Y [sig(y)] < ∞. +2. If Ex∼X[sig(x)] = Ey∼Y [sig(y)], then X and Y are equal in the distribution sense. In +other words, they have the same law. +B +The Signature-Wasserstein-1 metric +We first recall the definition of the Wasserstein-1 distance between distributions defined on +the same measurable space. +Definition 8 (Wasserstein-1 distance). Let Π(µ, ν) denote the set of all joint distributions +on the measurable space (X × X, F ⊗ F) whose marginals are respectively µ and ν. Then the +Wasserstein-1 distance is defined as +W1(µ, ν) := +inf +γ∈Π(µ,ν) E(x,y)∼γ[||x − y||1] +(15) +In practice the infimum in (15) is highly intractable. +Luckily for us, the Kantorovich- +Rubinstein duality (Villani, 2009) states that the Wasserstein-1 distance can be calculated +as +W1(µ, ν) = +sup +∥F∥L≤1 +� +Ex∼µ[F(x)] − Ex∼ν[F(x)] +� +(16) +where the supremum is over all the 1−Lipschitz functions F : X → R. The most common +approach is to approximate F by a parameterized family of functions that are 1−Lipschitz, +like in the Wasserstein-GAN approach (Arjovsky et al., 2017). +Now let Z be a random variable taking values on a space Z. Let Gθ : Z → X be a map +parameterized by θ. Consider the pushforward conditional distribution in X that is given by +Gθ(Z), which we denote as Pθ, and let Pr denote the distribution of the data. Then, our goal +is to minimize an approximation of the Wasserstein-1 distance between Pθ and Pr. +In the Wasserstein GAN approach, the optimization problem that one tries to solve is +min +θ∈Θ max +ψ∈Ψ +� +Ex∼Pr[Fψ(x)] − Ez∼Z[Fψ(Gθ(z))] +� +(17) +where {Fψ}ψ∈Ψ is a family of 1−Lipschitz functions. The training algorithm consists in +performing k steps on the parameters on the critic Fψ, and once a good enough approximation +20 + +of the Wasserstein-1 distance has been reached, perform one step on the parameters of the +generator Gθ. However, this method is very unstable and sensitive to the choice of the +hyperparameters. +Remember that, in our case, we are interested in approximating distributions in path space, +such that X = A1([0, T]; Rd). Let K ⊆ X denote the union of the supports of µ and ν. By +definition of (16), there exists a sequence Fn : K → R of functions that are 1−Lipschitz such +that they attain the supremum W1(µ, ν). +If we assume that K ⊂ X is a compact subset, by the universal non-linearity property +of the signature we have that, ∀ϵ > 0 and for each Fn there exists a linear functional +ℓn : T((Rd+1)) → R such that +sup +x∈K |Fn(x) − ℓn(sig(x))| ≤ ϵ +(18) +This implies that the supremum in (16) can be attained by a sequence of linear functionals +applied to the signature, and we have that (16) is equivalent to +sigW1(µ, ν) = sup +∥ℓ∥L≤1 +� +Ex∼µ[ℓ(sig(x))] − Ex∼ν[ℓ(sig(x))] +� +(19) += sup +∥ℓ∥L≤1 +ℓ +� +Ex∼µ[sig(x)] − Ex∼ν[sig(x)] +� +(20) +where the expected signature is well defined, since we assumed that both measures have +a compact support. Moreover, by using the truncated signatures and Corollary 2 we can +approximate (19) uniformly by +sigNW1(µ, ν) = sup +∥ℓ∥L≤1 +ℓ +� +Ex∼µ[sigN(x)] − Ex∼ν[sigN(x)] +� +(21) +where now the linear functionals ℓ are defined in the truncated tensor algebra T (N)(Rd+1). +Since the domain space is finite dimensional, we have that the Lipschitz constant of a linear +functional ℓ : T (N)(Rd+1) → R is simply defined as the norm of its coefficients as an euclidean +vector. +It turns out that when we choose this to be the L2 norm, we have that the optimization +problem (21) admits a closed-form solution (Ni et al., 2020, Lemma A.3), with the solution +being +sigNW1(µ, ν) = +���Ex∼µ[sigN(x)] − Ex∼ν[sigN(x)] +��� +2 +(22) +which is called the truncated Signature-Wasserstein-1 metric of order N. In the GAN setting, +we can approximate the expected signatures of the model and the data by performing a +Montecarlo simulation, i.e. by computing the empirical averages of the signatures. +21 + +C +The Conditional Signature-Wasserstein GAN approach +What if we have some knowledge we want to condition this distribution on? Let X = +{Xt}t∈[0,Tx] and Y = {Yt}t∈[0,Ty] be two stochastic processes taking values on +X = A1([0, Tx]; Rdx), +Y = A1([0, Ty]; Rdy) +(23) +respectively. Let p(X, Y ) denote their joint distribution. Then, given any known trajectory +x ∈ X, we are interested in approximating the conditional distribution p(Y |X = x), which +we will compactly denote by Y |x. +In practice, we cannot approximate the expected signature of Y |x by a Montecarlo procedure, +since for every x we are usually given only one sample. Therefore we must come up with a +different method to estimate Ey∼Y |x[sig(y)]. +Just like in supervised learning3, we can formulate this problem as trying to approximate a +map +f : +X +→ +P(Y) +x +�→ +Y |x +(24) +where P(Y) is the space of all stochastic processes taking values on Y. If we restrict ourselves +to family of stochastic processes with compact support, by Theorem 4 and Lemma 1 there +will exist a unique function ˆf : T((Rdx+1)) → T((Rdy+1)) such that +ˆf(sig(x)) = Ey∼Y |x[sig(y)] +(25) +where we wrote f(x) = Y |x. If we assume that the domain of f is a compact subset, we can +use the uniform convergence of the truncated signatures to approximate (25) by +ˆfN,M : +K ⊂ T (N)(Rdx+1) +→ +T (M)(Rdy+1) +sigN(x) +�→ +Ey∼Y |x[sigM(y)] +(26) +The image space of ˆfN,M is real and has finite dimension, so one can think of it as an euclidean +space. Therefore we can use the universal non-linearity of the signature to approximate ˆfN,M +by a linear functional ℓN,M, +ℓN,M : +K ⊂ T (N)(Rdx+1) +→ +T (M)(Rdy+1) +sigN(x) +�→ +Ey∼Y |x[sigM(y)] +(27) +In conclusion, we have reduced a non-linear modelling problem between two continuous +and infinite dimensional spaces in (24) to a linear problem between two discrete and finite +dimensional spaces. Moreover, the function ℓN,M represents a conditional expectation, and so +we can use the classical linear regression framework to approximate it. +This is, given some data {x(i), y(i)}n +i=1, we will assume that +sigM(y(i)) = W · sigN(x(i)) + ϵi +(28) +where +3The difference with respect to supervised learning is that in this case we are interested in the conditional +distribution, and not its expectation. +22 + +• sigM(y(i)) can be viewed as an (dy+1)M−1 +dy +dimensional real vector. +• sigN(x(i)) can be viewed as an (dx+1)N−1 +dx +dimensional real vector. +• W is a real matrix of dimension (dy+1)M−1 +dy +× (dx+1)N−1 +dx +. +• ϵi ∼ N(0, σ2I) is i.i.d. Gaussian noise, with I denoting the identity matrix. +and use any standard libraries to compute the weight matrix W. +In practice, we will proceed as follows. Let Z be a random variable taking values on a space +Z. Let Gθ : Z × X → Y be a map parameterized by θ. Then, for a known x ∈ X, we will +consider the pushforward conditional distribution in Y that is given by Gθ(Z, x). Just like in +the unconditional case, the expected signature of Gθ(Z, x) is estimated through a Montecarlo +procedure. +In order to estimate the expected signature of Y |x from the set of given data, we simply +perform the linear regression explained earlier, from which we obtain an estimate of (27), +that we denote by ˆℓ. Then we can obtain an approximation of Ey∼Y |x[sig(y)] by +ˆEy∼Y |x[sigM(y)] = ˆℓ(sigN(x)) +(29) +D +Experimental Details +The experimental details that were used in Section 4, such as the model architectures and +the selected hyperparameters, can be found in the following pages. +D.1 +Architectures for the resources cost analysis +LSTM Conditional WGAN The LSTMs of the conditional generator had 5 hidden layers +of width 32, with 5 dimensional random noise, and had 77,089 learnable parameters. The +LSTMs of the critic also had 5 hidden layers of width 32, with 76,609 learnable parameters. +LSTM Conditional Sig-WGAN The architecture of the generator was the same as the +one we just described for the LSTM Conditional WGAN generator. The signature transform +was truncated at depth 5 and 4, for the input and output paths, respectively. We also applied +the cumulative sum transformation4. Overall, the dimension of both truncated signatures +was 363 and 120, respectively. +Neural SDE Conditional Sig-WGAN The number of hyperparameters in a conditional +Neural SDE is slightly larger. Since it is a little bit hard to keep track of all of them, we will +mention them along the corresponding notation we used in Definition 1. +4Although the truncated signatures approximate arbitrarily well the signature transform, this does not +mean that some crucial information might be lost by not computing higher levels. In order to mitigate this, +usually some transformations are applied to the given stream, see Morrill et al., 2020. +23 + +The size dz of the Neural SDE was 92. The initial random noise size dv was 16 and the +network ξ2 +θ was formed by one hidden layer of size 32. The network ξ1 +θ was simply a linear +function. The initial condition of the SDE was formed by a random part k of size 8 and a +deterministic part dz − k of size 84. The diffusion gθ was chosen to be of general type. The +dimension of the Wiener process dw was 10. Both the drift fθ and diffusion gθ had a hidden +layer of width 32. The only activation function we used was the hyperbolic tangent, tanh. +The total number of learnable parameters was 79,273. +For the Neural SDE the signatures in the CSig-WGAN algorithm were of the same order +and with the same transformations as the LSTM Conditional Sig-WGAN model. +The +transformation φ(x) of the input paths x was set to be the signature transform, with the +same truncation order and transformation the ones used in the CSig-WGAN algorithm. +D.2 +Architectures and hyperparameters for the experiments +In this section we will detail the architectures and hyperparameters that were used for each +experiment in Section 4.2, which were selected by performing an informal grid search. +First we will list the ones that were common to all the datasets. +D.2.1 +Common hyperparameters +The optimizers that were used were as follows: for the LSTM CWGAN, following Arjovsky +et al., 2017 we used RMSprop. For the models trained with the CSig-WGAN algorithm, we +used Adam. The learning rate was always set to 10−3. +In all cases the signature transform was truncated at depth 5 and 4, for the input and +output paths, respectively. We also applied the cumulative sum transformation, and we +normalized each dimension so that it had zero mean and unit variance. Overall, the dimension +of both truncated signatures was 363 and 120, respectively. The transformation φ(x) of the +Neural SDEs was set to be the signature transform, with the same truncation order and +transformation the ones used in the CSig-WGAN algorithm. +D.2.2 +AR(p) dataset +The training set was formed by 14,900 pairs of time series data (x, y). The validation set was +formed by 2,400 samples. The test set was formed by 12,400 samples. We normalized the +data with the mean and standard deviation of the input streams in the training set. +The architectures and other hyperparameters for each model were as follows. +LSTM WGAN The LSTMs in the generator had 5 layers with hidden size equal to 32, +with 5 dimensional random noise. The LSTMs in the critic had 4 layers with hidden size 32. +The number of parameters of the generator was 77,089, while the number of parameters of +the critic was 59,713. +24 + +LSTM Sig-WGAN As we already explained, the generator was the same as in the LSTM +WGAN model. +Neural SDE Sig-WGAN The size dz of the Neural SDE was 48. The initial random noise +size dv was 16 and the network ξ2 +θ was formed by one hidden layer of size 32. The network ξ1 +θ +was simply a linear function. +The initial condition of the SDE was formed by a random part k of size 16 and a deterministic +part dz − k of size 32. The diffusion gθ was chosen to be of diagonal type, and therefore the +dimension of the Wiener process dw was the same as the size of the solution, 48. Both the +drift fθ and diffusion gθ had one hidden layer of width 84. The only activation function we +used was the hyperbolic tangent, tanh. Following Kidger, Foster, Li, Oberhauser, et al., 2021, +we also applied a final tanh to the vector fields. The total number of learnable parameters +was 29,161. +For the LSTM WGAN and the Neural SDE Sig-WGAN models, the batch size was set to +528. Due to memory constrains, for the LSTM Sig-WGAN the batch size was set to 228. +D.2.3 +Seattle Weather dataset +The train set was formed by 15,122 pairs of time series data (x, y). The validation set was +formed by 3,781 samples. The test set was formed by 6,468 samples. The test set was +extracted from a different time period than the train and validation sets (out-of-time samples). +We normalized the data with the mean and standard deviation of the input streams in the +training set. +The architectures and other hyperparameters for each model were as follows. +LSTM WGAN The LSTMs in the generator had 5 layers with hidden size equal to 32, +with 5 dimensional random noise. The LSTMs in the critic had 4 layers with hidden size 36. +The number of parameters of the generator was 77,089, while the number of parameters of +the critic was 75,241. +LSTM Sig-WGAN The generator was the same as in the LSTM WGAN model. +Neural SDE Sig-WGAN The architecture was the same as the one we detailed in Section +D.2.2, with the exception that we set the hidden size of the SDE to dz = 64. Just like in +the previous experiment we applied a final tanh to the vector fields of the SDE. The total +number of parameters was 35,113. +For the Neural SDE Sig-WGAN models, the batch size was set to 724. For the LSTM WGAN, +it was set to 528. Due to memory constrains, for the LSTM Sig-WGAN the batch size was +set to 228. +D.2.4 +Forex dataset +The train set was formed by 15,000 pairs of time series data (x, y). The validation set was +formed by 5,000 samples. The test set was formed by 10,000 samples. The test set was +25 + +extracted from a different time period than the train and validation sets (out-of-time samples). +As it is usually done in this kind of datasets, we applied the logarithm transformation to the +data. After that, we normalized the data with the mean and standard deviation of the input +streams in the training set. +The architectures and other hyperparameters for each model were as follows. +LSTM WGAN The architecture of the generator was the same as the one we detailed +in the Seattle Weather dataset experiment. However, in this case the architecture of both +LSTMs of the critic was set to hidden size equal to 32 and a number of layers equal to 5. +The number of parameters of the critic was 234,113. +LSTM Sig-WGAN The generator was the same as in the LSTM WGAN model. +Neural SDE Sig-WGAN The architecture was the same as the one we detailed in the +Seattle Weather dataset experiment. +For the LSTM WGAN and the Neural SDE Sig-WGAN, the batch size was set to 528. Due +to memory constrains, for the LSTM Sig-WGAN the batch size was set to 128. +D.2.5 +IBEX35 dataset +The train set was formed by 4,296 pairs of time series data (x, y). The validation set was +formed by 1,074 samples. The test set was formed by 1,944 samples. The test set was +extracted from a different time period than the train and validation sets (out-of-time samples). +We normalized the data with the mean and standard deviation of the input streams in the +training set. +The architectures and other hyperparameters for each model were as follows. +LSTM WGAN The LSTMs in the generator had 2 layers with hidden size equal to 64, +with 5 dimensional random noise. The LSTMs in the critic had 4 layers with hidden size 32. +The number of parameters of the generator was 101,953, while the number of parameters of +the critic was 59,713. +LSTM Sig-WGAN The generator was the same as in the LSTM WGAN model. +Neural SDE Sig-WGAN The size dz of the Neural SDE was 120. The initial random +noise size dv was 16 and the network ξ2 +θ was formed by one hidden layer of size 48. The +network ξ1 +θ was simply a linear function. +The initial condition of the SDE was formed by a random part k of size 56 and a deterministic +part dz − k of size 64. The diffusion gθ was chosen to be of diagonal type, and therefore the +dimension of the Wiener process dw was the same as the size of the solution, 64. Both the +drift fθ and diffusion gθ had one hidden layer of width 96. The only activation function we +used was the hyperbolic tangent, tanh. Just like in the previous experiments we applied a +final tanh to the vector fields of the SDE. The total number of learnable parameters was +73,489. +For the LSTM WGAN and the Neural SDE Sig-WGAN, the batch size was set to 1024. Due +to memory constrains, for the LSTM Sig-WGAN the batch size was set to 528. +26 + +E +Computing infrastructure +All experiments were run on a computer that had Windows 11 Home as the operative system, +equipped with an AMD Ryzen 7 5800X 8-Core Processor, 16GB of RAM and an Nvidia +GeForce RTX 3060 with 12GB of memory. +The main python libraries that were used are listed below: +• Pytorch 1.9.0+cu111 as the main deep learning framework (Paszke et al., 2019). +• Signatory 1.2.6, which provides differentiable computations of the signature on the +GPU (Kidger & Lyons, 2021). +• torchsde 0.2.5, which provides stochastic differential equation (SDE) solvers with GPU +support and efficient backpropagation (Li, 2020). +We highlight that, in order to use the Signatory package, one needs to have a Pytorch version +that is no older than 1.9.0. +27 + diff --git a/RtAzT4oBgHgl3EQfXPx9/content/tmp_files/load_file.txt b/RtAzT4oBgHgl3EQfXPx9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..48af1ae7b58b5876d84a57ba85cb730bf85cb13e --- /dev/null +++ b/RtAzT4oBgHgl3EQfXPx9/content/tmp_files/load_file.txt @@ -0,0 +1,992 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf,len=991 +page_content='Neural SDEs for Conditional Time Series Generation and the Signature-Wasserstein-1 metric Pere Díaz Lozano Departament de Matemàtiques Universitat Autònoma de Barcelona pere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='diaz@uab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='cat Toni Lozano Bagén Departament de Matemàtiques Universitat Autònoma de Barcelona tonilb@mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='uab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='cat Josep Vives Departament de Matemàtiques i Informàtica Universitat de Barcelona josep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='vives@ub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='edu January 5, 2023 Abstract (Conditional) Generative Adversarial Networks (GANs) have found great success in recent years, due to their ability to approximate (conditional) distributions over extremely high dimensional spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' However, they are highly unstable and computa- tionally expensive to train, especially in the time series setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Recently, it has been proposed the use of a key object in rough path theory, called the signature of a path, which is able to convert the min-max formulation given by the (conditional) GAN framework into a classical minimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' However, this method is extremely expensive in terms of memory cost, sometimes even becoming prohibitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' To overcome this, we propose the use of Conditional Neural Stochastic Differential Equations, which have a constant memory cost as a function of depth, being more memory efficient than traditional deep learning architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' We empirically test that this proposed model is more efficient than other classical approaches, both in terms of memory cost and computational time, and that it usually outperforms them in terms of performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Keywords: conditional generative modelling, neural networks, expected signature, rough path theory, Wasserstein generative adversarial networks, neural stochastic differential equa- tions arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='01315v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='ML] 3 Jan 2023 1 Introduction The simplest approach to perform time series forecasting is to only be interested in a determin- istic response, which is usually thought of as the mean of possible outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Consequently, the model is unable to report any inherent uncertainty, which is a major shortcoming for most real world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Recent work has ought to solve this by turning their attention to Generative Adversarial Networks (GANs) (Arjovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', 2017, Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Their basic idea is to train two networks against each other: The generator: it generates samples from a distribution by sending random noise through a parameterized model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The adversarial network: its job is to approximate a loss function between the real data distribution and the distribution produced by the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The training algorithm consists in training the adversarial network for a certain numbers of steps, and once a good enough approximation of the loss function we are interested in is reached, then perform one step on the parameters of the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The original GAN formulation was set such that the adversarial network approximated the Jensen Shannon divergence (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' However, in our case we will focus on the Wasserstein GAN formulation, where the adversarial network (called the critic) aims at approximating the Wasserstein-1 distance (Arjovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Although GANs have had great success in approximating probability measures over extremely high dimensional spaces, especially in Computer Vision, they are very unstable to train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Moreover, the fact that one needs to train an adversarial network before performing one step on the generator means that the training time is considerably large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='1 Signature-Wasserstein-1 metric A more efficient approach would be to approximate the loss function in a closed form way, without the need of an iterative method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' In this direction, Ni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', 2021 proposed the use of the signature transform, a fundamental object from rough path theory which captures many of the most important analytic and geometric properties of a given path1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' By using it, one is able to derive an expression that is able to approximate uniformly well the Wasserstein-1 distance between two distributions on path space in a closed form non-parametric way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' In doing so, all of the drawbacks that are inherent to the (Wasserstein) GAN formulation are overcome, while still maintaining its strong theoretical guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' This new framework is called the Signature-Wasserstein GAN (SigWGAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' 1Although in practice we are usually given a set of discrete streams of data, we can always embed them into the set of continuous paths by performing some kind of interpolation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Therefore, we will not really differentiate between these two concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' 1 In Ni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', 2020, the authors proposed the Conditional Signature-Wasserstein GAN (SigCW- GAN), which is a modification of the SigWGAN to learn instead conditional distributions, as the ones we are interested in when doing non-deterministic time series forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' However, since this algorithm needs to perform a Montecarlo procedure for each data sample, the memory cost is dramatically increased, sometimes even becoming prohibitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' They elude this by assuming that the distribution of the observed time series has an autoregressive structure, with the next observed value depending only on the previous q > 0, with q relatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='2 Contributions In order to overcome this increase in the memory cost, we propose the use of Neural Stochastic Differential Equations (Neural SDEs) (see Kidger, Foster, Li, Oberhauser, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', 2021, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', 2020), which are essentially generative models formed by classical stochastic differential equations whose vector fields are parameterized by neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The main advantage over traditional deep learning architectures is the fact that one is able to backpropagate through a Neural SDE without the need to store the intermediate quantities produced in the forward pass, being more memory efficient than traditional deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' By encoding the path we want to condition on, and by making the initial condition of the Neural SDE depend on this codification, we introduce what we call Conditional Neural Stochastic Differential Equations (CNSDEs), which produce conditional distributions in path space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' When using all of the above, we are able to formulate a model and a training algorithm for approximating conditional distributions on path space, which is both mathematically well founded and practically more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Furthermore, we will test it against other more traditional baselines, concluding that in most cases it outperforms them according to several metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' 2 Background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='1 The Signature-Wasserstein-1 metric Let sig(x), sigN(x) denote the signature and the truncated signature of order N of a continuous path x, respectively, both with the basepoint and time augmentations (see Morrill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Let µ, ν be two distributions in path space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The Kantorovich-Rubinstein duality states that the Wasserstein-1 distance can be calculated as W1(µ, ν) = sup ∥F∥L≤1 � Ex∼µ[F(x)] − Ex∼ν[F(x)] � where the supremum is over all the 1−Lipschitz functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' 2 In Ni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', 2021, Ni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', 2020 the authors use the signature transform and its universal non-linearity to derive a closed form expression that uniformly approximates the Wasserstein-1 distance, sigNW1(µ, ν) = ���Ex∼µ[sigN(x)] − Ex∼ν[sigN(x)] ��� 2 (1) where ∥·∥2 denotes the L2 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' This approximation is called the truncated Signature- Wasserstein-1 metric of order N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Sections A and B in the supplementary material provide an in depth derivation of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='2 The Conditional Signature-Wasserstein GAN algorithm In the unconditional case, whenever we intend to compute (1) from a set of given data, we can simply approximate both expected signatures by their empirical average, by performing a Montecarlo simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' However, if instead we are interested in approximating conditional distributions, then one is not able to perform a Montecarlo simulation to approximate the expected signatures of the conditional distributions given by the data, since most of the times we are only handled one sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' By using many of the properties of the signature, one is able to prove that the conditional expected truncated signature can be approximated arbitrarily well by applying a linear map on the truncated signature of the conditioning path x, ˆEy∼Y |x[sigM(y)] = ˆℓ(sigN(x)) (2) which can be approximated from the data by standard linear regression techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' An in depth derivation of (2) is given in Section C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The resulting training algorithm is called the Conditional Signature-Wasserstein GAN (SigCW- GAN) (see Ni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', 2020, Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Figure 1: Flowchart of the SigCWGAN algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' 3 Input path a Ey~Yla[sigM(y)] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' e(sigN (α)) SigN-Wi metric Z noise Conditional Generator Ey~Go(Z,a)[sig M (y)] distribution Ge(z,α) Monte Carlo2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='1 Disadvantages of the SigCWGAN algorithm One of the main drawbacks of the SigCWGAN algorithm is the Montecarlo procedure needed to estimate the expected signature of the conditional generator for each sample in the minibatch, which considerably increases the memory cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' This is in contrast to the general conditional GAN setting, where both the generator and the critic simply take as input the corresponding sample x (Mirza & Osindero, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Therefore, the conditional Wasserstein GAN algorithm (CWGAN) has the same memory cost as the unconditional WGAN approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' This means that the SigCWGAN algorithm consumes much more memory that the traditional CWGAN approach, and as a consequence sometimes we need to decrease the batch size or the complexity of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' As we will see in the following pages, this can be remedied by using a Neural Stochastic Differential Equation as a generator, which are much more memory efficient to train than traditional neural networks architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='3 Neural Stochastic Differential Equations Neural Stochastic Differential Equations (Neural SDE) are models that arise from the union of stochastic differential equations and neural networks, two of the biggest paradigms in mathematical modelling, resulting in generative models that produce distributions in path space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The basic structure of a Neural SDE is Z0 ∼ ξθ(V ) dZt = fθ(t, Zt)dt + gθ(t, Zt) ◦ dWt Yt = αθZt + βθ with V ∼ N(0, Idv) drawn from a dv−dimensional standard normal distribution, ξθ, fθ, gθ being neural networks and αθ, βθ being matrices of learnable weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Wt denotes the Wiener process, and the ◦ indicates that the integration is in the Stratonovich sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Z represents the hidden state of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' If it was the output, then the resulting stochastic process would satisfy a Markov property, which does not need to be true in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' This is the reason for the final readout linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' As we already said, the solution to a stochastic differential equation is a distribution in path space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' However, just like with ODEs, computing it in an analytical form is almost always impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Nonetheless, one can sample from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' This is what a numerical SDE solver does, returning a sampled path evaluated at a set of discrete time locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Thus a Neural SDE fits the GAN setting, since evaluating its density is not possible and it is able to generate samples by sending random noise (in the form of the initial condition and the Wiener process) through a parameterized model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='1 Backpropagation through a Neural SDE If we intend to fit a Neural SDE to some data, we need an algorithm to compute gradients with respect to its parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' We will briefly summarize two ways to do this: Discretize-then-optimize: The most straightforward way consists in performing the usual backpropagation, differentiating through the internal operations of the differential equation solver, and therefore needing to store them in the forward pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' In the literature this is commonly referred to as discretize-then-optimize, because we are directly optimizing the discretization we are using in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' If we denote by H the memory cost of recording the operations of one solver step, and by N the number of steps, then the discretize-then-optimize method consumes about O(HN) memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Notice that this is the memory cost of traditional deep learning models, such as Recurrent Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Reversible solvers: Alternatively, if we use a reversible solver, we dramatically reduce the memory cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Consider a differential equation solver, which in the forward pass iteratively computes the next step from the previous one, (zti, αti) �→ (zti+1, αti+1), (3) where {zti}n i=1 is the numerical approximation of the solution to some differential equation, and αti represents the intermediate quantities that the solver needs to store at step i in order to compute the solution at the next step i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Then, a solver is said to be algebraically reversible if it can compute the previous step from the next step, (zti+1, αti+1) �→ (zti, αti), (4) by using a closed-form expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Notice how, in this case, the intermediate values (zti, αti) do not need to be stored in memory, and they can be recomputed in the backward pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' In this case, the memory cost is only O(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The full algorithm, along with the only known such SDE solver (called the reversible Heun method), can be found in Kidger, Foster, Li, and Lyons, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' 3 Conditional Neural Stochastic Differential Equations In our case we are interested in generating distributions that are conditioned on some input path x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' We propose to condition a Neural SDE by making the initial condition of the SDE depend on the path we want to condition on, following an encoder-decoder structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Definition 1 (Conditional Neural Stochastic Differential Equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Let ξθ : Rdh → Rdz fθ : [0, T] × Rdz → Rdz gθ : [0, T] × Rdz → Rdz×dw 5 be feedforward neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Let φ : T S([0, τ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Rdx) → Rdh be a continuous map (with or without learnable parameters), with T S([0, τ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Rdx) being the space of time series with timestamps in [0, τ] and dx−dimensional observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Then we define a Conditional Neural Stochastic Differential Equation (CNSDE) as h = φ(x) Z0 = ξθ(h) dZt = fθ(t, Zt)dt + gθ(t, Zt) ◦ dWt Yt = αθZt + βθ where αθ ∈ Rdy×dz and βθ ∈ Rdy are matrices of learnable weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The following is an overview of a CNSDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' An input path x (red) is fed into the model, which determines the initial condition of the SDE z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Then a trajectory w is sampled from the Wiener process (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' All of this is fed to the differential equation solver, which gives us the solution z (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' After that we apply a final readout linearity, which produces the output of the model y (purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' As a summary, y is a sample from a distribution in path space that is conditioned on a given sample x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Whenever we change x, this distribution changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Figure 2: Overview of a Conditional Neural SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Notice how the initial condition of the SDE is completely determined (once the parameters are fixed) by the input path x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' An alternative approach is to add a random component to the initial condition Z0, which usually enforces diversity and improves learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' This can be simply done by setting the initial condition to be h = φ(x) (5) Z0 ∼ [ξ1 θ(h), ξ2 θ(V )] (6) with ξ1 θ : Rdh → Rdz−k and ξ2 θ : Rdv → Rk being neural networks and V ∼ N(0, Idv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The larger the size of k with respect to dz, the more we will be enforcing diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' 6 M~m y=αe+4 Empirical Analysis The goal of this section is to compare the performance of the SigCWGAN algorithm and the CNSDE against some more traditional approaches which will serve as a baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' All the code is available in the GitHub repository https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='com/pere98diaz/Neural-SDEs-for- Conditional-Time-Series-Generation-and-the-Signature-Wasserstein-1-metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' We will compare the performance of three models: LSTM Conditional Wasserstein GAN: the model that will serve as a pure baseline is based on Long short-term memory (LSTM) networks (Hochreiter & Schmidhuber, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' It is trained by using the Conditional Wasserstein GAN algorithm, with the critic being also formed by LSTMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' LSTM Conditional Signature-Wasserstein GAN: the second model is formed by the same conditional generator as the LSTM CWGAN model, but using the truncated Signature- Wasserstein-1 metric to approximate the Wasserstein-1 distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' To train the model we use the SigCWGAN algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Neural SDE Conditional Signature-Wasserstein GAN: the third model is formed by a Conditional Neural Stochastic Differential Equation as the generator, and uses the truncated Signature-Wasserstein-1 metric to approximate the Wasserstein-1 distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' We also use the SigCWGAN algorithm for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='1 Resources cost In this section we will be comparing the three models defined earlier in terms of two key resources: maximum memory allocated on the GPU (left hand side) and computational time required to perform one step on the generator (right hand side).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The architectures of the three models are chosen so that they have a similar number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Figure 3: Memory consumption (MB) and time per step (s) for the three models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' We can see how, in terms of memory cost, the LSTM trained with the SigCWGAN algorithm is by far the most expensive one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' This is mainly because of the Montecarlo procedure we need to perform for each sample on the minibatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' However, notice that when using the 7 Memory cost Time per step 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='5 DO 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='5 step DO+ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='0 per!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='5 4000 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='D 2400 25 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='D LSTM CWGAN LSTM SigCWGAN CNSDE SigCWGAN LSTM CWGAN LSTM SigCWGAN CNSDE SigCWGANCNSDE as the conditional generator, the memory drops considerably, due to the use of a reversible solver to perform backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Unsurprisingly, the LSTM trained with the CWGAN algorithm is the cheapest one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' In terms of time per generator step, the most expensive model is the LSTM trained with the CWGAN algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' This is mainly because of the fact that we need to perform k steps on the parameters of the critic before performing one step on the parameters of the generator (in our case, we picked the standard default value, which is k = 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The models trained with the SigCWGAN algorithm approximate the Wasserstein-1 loss in a closed form non-parametric way, which is the reason why it takes much less time to perform one generator step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' In conclusion, the most balanced model out of the three is the CNSDE trained with the SigCWGAN algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='2 Experiments We performed experiments on four one dimensional different datasets, which will be described next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Each of the three models were trained three times, and we measured their performance in terms of the following statistics, all evaluated in an out-of-time test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Classification error: We train an LSTM whose task is to, given a pair of input/output streams, classify whether the output stream is real or generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The architecture of this classifier is the same for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The worse the performance of this classifier, the better the generative model is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Higher order Signature-Wasserstein-1 metric: Similarly to the SigCWGAN algorithm, we compute the L2-distance between the predicted and generated expected signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' However, in this case we truncate the signatures at higher levels than the ones we used during training: depth 6 for the input paths and depth 5 for the output paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Unordered Wasserstein-1 metric: We compare the Wasserstein-1 distance between the real one dimensional distributions that are given by: a) taking the data points yt from the output streams, without considering that they are ordered in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' b) taking the differences between the data points in the output streams and the last value from the input stream, yt − xT, also without considering that they are ordered in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' c) For each stream, taking the largest difference given by the procedure in b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' d) For each stream, taking the smallest difference given by the procedure in b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Extreme values metric: Consider the empirical distribution given by the procedure we detailed in the Unordered Wasserstein-1 metric c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Let q indicate the value of a very high percentile of it, for example 95%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Then we estimate the probability, conditioned on the input path x, of producing a path y such that maxt yt − xT is equal or over q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' In some cases we will instead consider percentage increases, as maxt(yt − xT)/xT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Just like we defined a way of measuring how good the model can detect a high possibility of an extremely large increment, we can do the same for an extremely value decrease, by 8 considering mint yt − xT instead (or mint(yt − xT)/xT) and setting the percentile to be very low, like 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' We should remark that evaluating the performance of a conditional generative model, especially in the time series framework, is very challenging, and none of the above statistics should be considered as ground truth metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Moreover, most of the time the metric we are interested in depends on the problem we have at hand, and the purpose and use we want to give to that model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='1 AR(5) data The first dataset that we used was simulated from an autoregressive model of order p = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' We considered the problem of predicting the next 40 steps given the previous 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The training time was set to a maximum of 2 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' However, for the models using the SigCWGAN algorithm one is able to define an early stopping criteria, which was set to 1,000 steps without improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Moreover, at the end of training one can just keep the parameters that gave the best loss on a validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The next table indicates some general information on training, like the maximum memory allocated, the training time or the number of steps that were performed on the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The last column indicates the objective function that was used for training in the SigCWGAN models evaluated in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' For completeness, we also show it for the LSTM CWGAN, although it was not set to explicitly optimize it at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Memory (MB) Time (h) Steps Sig-W1 loss LSTM CWGAN 1945 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='000 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='000 566 ± 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='578 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='297 LSTM SigCWGAN 9931 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='712 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='499 8789 ± 949 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='804 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='048 NSDE SigCWGAN 3764 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='555 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='386 16839 ± 1673 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='855 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='095 Table 1: Some general information on the training process, AR(5) dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Next we show the Area Under the Curve (AUC) and the accuracy obtained from training an LSTM to tell apart real from fake data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' We can clearly see how the LSTM CWGAN outperforms the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' This is not a surprise at all, since in the GAN framework the generator is directly competing against another network whose job is precisely to distinguish real data from generated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The best model according to each metric is always marked as bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' AUC Accuracy LSTM CWGAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='747 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='144 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='685 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='121 LSTM SigCWGAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='866 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='775 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='043 NSDE SigCWGAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='959 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='873 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='031 Table 2: Classification error, AR(5) dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' 9 HO Sig-W1 metric EV+ AUC EV− AUC LSTM CWGAN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='412 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='361 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='831 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='098 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='834 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='130 LSTM SigCWGAN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='092 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='958 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='960 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='035 NSDE SigCWGAN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='074 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='988 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='982 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='013 Table 3: The Higher order Signature-Wasserstein-1 metric and both the Extreme values metrics, AR(5) dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The selected percentiles for the EV+ and EV− were 99% and 1%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' W1-a W1-b W1-c W1-d LSTM CWGAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='092 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='127 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='061 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='216 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='296 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='236 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='301 LSTM SigCWGAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='086 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='056 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='155 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='089 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='135 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='059 NSDE SigCWGAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='036 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='017 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='156 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='165 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='061 Table 4: The four unordered Wasserstein-1 metrics, AR(5) dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' In conclusion, we see in Table 1 that the Neural SDE did a better job minimizing the Sig-W1 loss function than both LSTMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' This did not translate into a better performance in terms of the Classification error in Table 2, meaning that probably the Signature-Wasserstein-1 metric failed to produce a good enough approximation of the codification of the conditioned stochastic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' However, in terms of the rest of the metrics, the Neural SDE outperforms the other models, implying that it was able to encode many of its most important geometric properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='2 Seattle weather The second dataset was formed by daily observations of the maximum temperature reached in Seattle from 1948 to 20172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Our task will be to, given the last 60 days, predict the next 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The training time was set to a maximum of 3 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' For the models using the SigCWGAN we set an early stopping criteria of 1,000 steps without improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Memory (MB) Time (h) Steps Sig-W1 loss LSTM CWGAN 2055 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='000 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='000 965 ± 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='944 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='360 LSTM SigCWGAN 8881 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='937 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='201 7501 ± 1639 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='119 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='009 NSDE SigCWGAN 4696 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='317 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='460 4834 ± 1626 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='410 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='067 Table 5: Some general information on the training procedure, Seattle Weather dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' 2The dataset can be found in https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='com/datasets/rtatman/did-it-rain-in-seattle-19482017 10 AUC Accuracy LSTM CWGAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='617 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='573 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='033 LSTM SigCWGAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='715 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='114 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='652 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='087 NSDE SigCWGAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='732 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='660 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='031 Table 6: Classification error, Seattle Weather dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' HO Sig-W1 EV+ AUC EV− AUC LSTM CWGAN 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='014 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='514 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='577 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='047 LSTM SigCWGAN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='465 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='265 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='808 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='888 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='046 NSDE SigCWGAN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='032 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='082 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='905 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='940 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='003 Table 7: The Signature-Wasserstein-1 metric and both the Extreme values metrics, Seat- tle Weather dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The selected percentiles for the EV+ and EV− were 95% and 5%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' W1-a W1-b W1-c W1-d LSTM CWGAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='061 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='155 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='251 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='175 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='018 LSTM SigCWGAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='060 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='035 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='094 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='144 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='072 NSDE SigCWGAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='047 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='028 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='062 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='055 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='008 Table 8: The four unordered Wasserstein-1 metrics, Seattle Weather dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The conclusions are pretty much similar to the ones we obtained for the AR(5) dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' In Table 5 we see that the Neural SDE did a way better job at minimizing the CSig-W loss function than the LSTMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Since this did not translate to the Classification error performance showed in Table 6, this means that the Signature-Wasserstein-1 metric did not do a great job at codifying the conditional stochastic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' However, since the Neural SDE outperforms the other models in terms of the rest of the metrics, we also conclude that it succeeded in capturing many important geometric properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' We especially highlight the results showed in Table 7, where we can see how the LSTM trained with the WGAN algorithm completely failed to perform well in terms of detecting extreme values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' This is in contrast to the models trained with the SigCWGAN method, which did a very good job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='3 Forex The third dataset was a Forex time series formed by observations of the bid price between the Euro and the Dollar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' More specifically, at each timestamp it indicates the highest price a buyer will pay, in Dollars, to buy one Euro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The observations are spanned over weeks 19 and 20 of 2020, and are irregularly spaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The NSDE SigCWGAN model is able to work with irregularly sampled data, since both the conditioner and the loss function are based on the signature transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' However, this is not 11 the case for the other two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' In order to be able to compare them, we aggregated the data by computing the mean in each 30 seconds interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Our task will be to, given the last 80 observations, predict the next 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The training time was set to a maximum of 4 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' For the models using the SigCWGAN algorithm, we set an early stopping criteria of 1,000 steps without improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Memory (MB) Time (h) Steps Sig-W1 loss LSTM CWGAN 5818 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='000 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='000 718 ± 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='860 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='947 LSTM SigCWGAN 9209 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='555 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='252 10417 ± 1664 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='491 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='002 NSDE SigCWGAN 8697 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='653 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='236 3001 ± 433 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='049 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='019 Table 9: Some general information on the training procedure, Forex dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' AUC Accuracy LSTM CWGAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='962 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='087 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='932 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='097 LSTM SigCWGAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='838 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='768 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='118 NSDE SigCWGAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='630 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='093 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='587 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='060 Table 10: Classification error, Forex dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' HO Sig-W1 EV+ AUC EV− AUC LSTM CWGAN 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='614 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='346 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='500 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='510 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='017 LSTM SigCWGAN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='010 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='488 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='499 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='005 NSDE SigCWGAN 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='704 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='077 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='596 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='544 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='013 Table 11: The Signature-Wasserstein-1 metric and both the Extreme values metrics, Forex dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The selected percentiles for the EV+ and EV− were 90% and 10%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' W1-a W1-b W1-c W1-d LSTM CWGAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='103 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='107 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='046 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='053 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='146 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='052 LSTM SigCWGAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='012 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='020 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='029 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='037 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='007 NSDE SigCWGAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='013 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='019 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='028 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='015 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='002 Table 12: The four unordered Wasserstein-1 metrics, Forex dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Notice how, in contrast to the last two experiments, the NSDE SigCWGAN model clearly outperformed the rest of the models in terms of Classification error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' We theorize that this is due to the increment, in terms of length, of both the input and output streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' However, further experiments should be conducted to test whether this is indeed the true reason or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' In this problem we perhaps could be especially interested in knowing whether, given a known path x, there will be a large increment or reduction in a fixed time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' This is tested with the Extreme values metric, and the results of each model are shown in Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' We can see how the NSDE based model also outperforms the rest in terms of these metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='4 IBEX35 The final dataset was formed by the daily return (between 1993 and 2022) of the IBEX 35 (IBerian IndEX), which is the benchmark stock market index of the Bolsa de Madrid, Spain’s principal stock exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Our task will be to, given the last 30 observations, predict the next 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The training time was set to a maximum of 2 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' For the models using the SigCWGAN algorithm, we set an early stopping criteria of 1,000 steps without improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Memory (MB) Time (h) Steps Sig-W1 loss LSTM CWGAN 1786 2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='000 1283 ± 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='837 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='297 LSTM SigCWGAN 9201 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='442 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='295 1334 ± 2919 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='788 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='020 NSDE SigCWGAN 5825 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='647 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='423 6917 ± 1773 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='811 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='016 Table 13: Some general information on the training procedure, IBEX35 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' AUC Accuracy LSTM CWGAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='844 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='765 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='104 LSTM SigCWGAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='661 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='066 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='618 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='052 NSDE SigCWGAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='588 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='074 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='562 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='058 Table 14: Classification error, IBEX35 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' HO Sig-W1 metric EV+ AUC EV− AUC LSTM CWGAN 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='910 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='231 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='776 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='507 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='118 LSTM SigCWGAN 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='606 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='867 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='642 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='048 NSDE SigCWGAN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='301 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='367 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='886 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='687 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='012 Table 15: The Signature-Wasserstein-1 metric and both the Extreme values metrics, IBEX35 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The selected percentiles for the EV+ and EV− were 95% and 5%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' It is interesting to mention that, in terms of the Extreme values metric, the performance of the models considerably drops during the COVID-19 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' For example, for the Neural SDE the results of the EV− AUC in the period 2015-2019 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='758 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='033, while in the period 2020-2022 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='633 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' W1-a W1-b W1-c W1-d LSTM CWGAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='025 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='021 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='041 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='023 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='013 LSTM SigCWGAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='014 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='022 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='027 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='025 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='001 NSDE SigCWGAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='014 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='017 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='029 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='016 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='002 Table 16: The four unordered Wasserstein-1 metrics, IBEX35 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' We can conclude that in this case the NSDE SigCWGAN clearly outperformed the rest of the models in terms of all metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' 13 5 Conclusion In this paper, we have proposed the use of a Conditional Neural Stochastic Differential Equation as a conditional generator in the SigCWGAN algorithm, which offsets the great increase in terms of memory cost produced by the Montecarlo procedure needed for every sample in the minibatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' We then tested in practice what was the gain and loss, in terms of computational time and memory cost, of first replacing the traditional WGAN algorithm with the SigCWGAN method, and then the traditional LSTM generator with a Neural SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' We clearly showed that Neural SDEs trained with the SigCWGAN algorithm were the most balanced in terms of both resources cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Finally, we compared their performance in four experiments with four different datasets, which allowed us to see that, in most cases, the Neural SDEs captured better some of the properties of the real datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Acknowledgements The research of Josep Vives is partially supported by Spanish grant PID2020-118339GB-100 (2021-2024).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Declarations of Interest All authors report no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The authors alone are responsible for the content and writing of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' References Arjovsky, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', Chintala, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', & Bottou, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Wasserstein generative adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' In D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Precup & Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='05421.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Ni, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', Szpruch, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', Wiese, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', Liao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', & Xiao, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Conditional sig-wasserstein gans for time series generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' SSRN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Paszke, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', Gross, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', Massa, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', Lerer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', Bradbury, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', Chanan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', Killeen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', Lin, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', Gimelshein, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', Antiga, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', Desmaison, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', Kopf, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', Yang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', DeVito, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', Raison, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', Tejani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', Chilamkurthy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', Steiner, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', Fang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Chintala, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Pytorch: An imperative style, high-performance deep learning library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' In H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Wallach, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Larochelle, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Beygelzimer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' d’Alché-Buc, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Fox, & R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Garnett (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' ), Advances in neural information processing systems 32 (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' 8024–8035).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' 15 Villani, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Optimal transport: Old and new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Grundlehren der mathematischen Wis- senschaften.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Springer, Berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' 16 Supplementary Material A The signature transform For completeness, we will give an elementary introduction to the signature transform, and we will briefly see many of the properties that make it such a convenient transformation in machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='1 Paths of bounded variation Definition 2 (p−variation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Let p ≥ 1 be a real number, and ∥·∥ any norm on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Let x : [0, T] → Rd be a continuous path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The p−variation of x is defined as ∥x∥p = � � sup D n−1 � i=0 ||xti+1 − xti||p � � 1/p (7) where the supremum is taken over the set of partitions of [0, T], denoted as D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The space of d−dimensional continuous paths of finite p−variation will be denoted as Vp([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' We will equip Vp([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Rd) with the following norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Definition 3 (p−variation norm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The p−variation norm of a path x ∈ Vp([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Rd) is defined as ∥x∥p−var = ∥x∥p + sup t∈[0,T] ∥xt∥ (8) where ∥·∥ is any norm in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' For simplicity, we will only work with paths of finite 1−variation, which are called of bounded variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The reason for this is that in practice we will always be given a discrete stream of data, which can be converted into a continuous path by performing some interpolation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The most common one is linear interpolation (with the resulting paths being of bounded variation), since then the signature is very easy to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' This is what the Signatory package (Kidger & Lyons, 2021) does, providing differentiable computations of the signature on the GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Moreover, to define a signature of a general path of finite p−variation, we would need to introduce many new concepts of rough path theory, which is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='2 The tensor algebra Definition 4 (Tensor algebra of Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Let (Rd)⊗k denote the kth tensor power of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' By convention, (Rd)⊗0 = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' We define the extended tensor algebra of Rd as T((Rd)) = {a = (a0, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=') | ∀k ≥ 0, ak ∈ (Rd)⊗k} 17 We also define the truncated tensor algebra of order N of Rd, which is a linear subspace of T((Rd)), as T (N)(Rd) = {a = (a0, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', aN) | ∀N ≥ k ≥ 0, ak ∈ (Rd)⊗k} Note that T (N)(Rd) is a real vector space of dimension N � k=0 dk = � � � N + 1 if d = 1 dN+1−1 d−1 if d > 1 (9) We will equip T((Rd)) with an admissible norm, defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' We say that the extended tensor algebra T((Rd)) is endowed with an admissible norm ∥·∥ if the following conditions hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' For each k ≥ 1, the symmetric group Sk acts by isometry on (Rd)⊗k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' ∥σv∥ = ∥v∥ , ∀v ∈ (Rd)⊗k, ∀σ ∈ Sk (10) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The tensor product has norm 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' ∀n, m ≥ 1, ∥v ⊗ w∥ ≤ ∥v∥ ∥w∥ , ∀v ∈ (Rd)⊗n, w ∈ (Rd)⊗m (11) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='3 The signature of a path The signature transform provides a way of encoding any continuous path into an infinite sequence of statistics, which has many properties that make it a very desirable transformation in machine learning (Chevyrev & Kormilitzin, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Definition 6 (Signature of a path).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' Let x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=', xd) : [0, T] → Rd be a continuous path of bounded variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfXPx9/content/2301.01315v1.pdf'} +page_content=' The signature of x is defined as the infinite collection of iterated integrals sig(x) = � � � · · � 0 0. +For comparison, Fig. 5 also contains empirical results ob- +tained with standard models using a β-law or double-β-law to +describe �(r) from Hamann et al. (2006, 2019); Hainich et al. +(2015); Shenar et al. (2016, 2019). Illustrating the well-known +“Wolf-Rayet radius problem” (e.g., Grassitelli et al. 2018), most +empirically derived temperatures are located at T∗ values that +seem to be too cool for their mass-loss rate. As demonstrated by +Gräfener & Hamann (2005) and Sander et al. (2020), the critical +radii inferred from dynamically-consistent atmosphere models +are much smaller than those obtained by a standard β-law due +to the opacities of the “hot iron bump” that enable the launch of +a supersonic wind already at deeper layers. Moreover, the com- +parison in the ˙M-Teff(τcrit) plane is not ideal as the empirically +derived values of ˙M depend on distances which are still uncer- +tain for some Galactic targets, but as we see below when dis- +cussing the transformed mass-loss rate, this is not a major issue +here. An inspection of the model sequences indicates a clear shift +for different mass regimes. Thus, one could argue that the whole +plane in Fig. 5 could be covered if we would calculate further +sequences for lower masses. However, lower masses are most +likely not the solution and discrepancies remain even when ad- +justing the mass-loss rates for stars with different luminosities in +Sect. D. +3.1. Mass loss versus different temperature scales +To discuss the different temperature scales in WR winds and +their scaling with ˙M, we take a closer look at an individual model +sequence. In Fig. 6, we plot the mass-loss rate ˙M for the se- +quence of 20 M⊙ models without hydrogen as functions of differ- +ent temperature definitions, namely (i) the effective temperature +T2/3 at a Rosseland optical depth of τR = 2/3 (thick red dashed +line), often simply denoted as Teff in the literature; (ii) the ef- +Teff(τR = 2/3) +T∗(τR,c = 20) +Teff(τcrit) +Te(Rcrit) +Grassitelli et al. (2018) +Teff(τs) +AS +-6 +-5 +-4 +20 +60 +100 +140 +180 +220 +T [kK] +log( ˙M [M⊙ yr−1]) +Fig. 7. Same as Fig. 6, but for a model sequence with log L/L⊙ = 5.475, +M = 15 M⊙, and XH = 0. Contrary to the situation in Fig. 6 for a 20 M⊙ +star, there is an abrupt breakdown of solutions beyond a minimum tem- +perature and a maximum ˙M. For the T2/3− and Teff(τcrit)-scales these +points are marked with a vertical line attached to a gray-hatched area. +fective temperature T∗ commonly used in the model setup for +PoWR models, defined at a Rosseland continuum optical depth +of τR,cont = 20; (iii) the effective temperature at the critical point, +denoted Teff(τcrit), Teff(Rcrit), or simply Teff,crit; and (iv) the elec- +tron temperature Te at the critical point (thin green dotted line). +In contrast to the first three temperatures, Te is not an effective +temperature. In the deeper layers of the atmosphere where the +deviations from LTE become negligible, Te aligns with the gen- +eral temperature T(r) defined in (1D) stellar structure models. +To further illustrate the effect of the two different nonmonotonic +�(r)-treatments, we plot the more accurate method with posterior +interpolation of the velocity field in strong colors while the sim- +ple method ignoring negative gradients is drawn in lighter shades +of the same line style. Beyond a certain temperature, there are no +more deceleration regions and thus both curves agree. +Except for the regimes with highest mass-loss ( ˙M +> +10−4 M⊙ yr−1 in Fig. 6), the values of T∗ and Teff,crit closely align. +This does not imply that τcrit has to correspond to τR,cont ≈ 20, +but that the locations of the radii corresponding to τR,cont = 20 +and τcrit are close enough to yield similar effective temperatures. +The alignment between T∗ and Teff,crit at lower ˙M is fulfilled in +all of our model sequences (see Figs. 7 and E.3 for further exam- +ples) and allows us to discuss the physically more meaningful +temperatures at the critical point (i.e., the launch of the wind) in- +stead of the slightly more technical T∗. For higher ˙M, τcrit moves +inward and eventually surpasses τR,cont = 20, explaining the de- +viation of the curves for the highest mass-loss rates. However, +these cases require models very close to the Eddington limit. +The growing difference between the effective temperature +at the launch of the wind (Teff,crit) and T2/3 with increasing ˙M +shows the “extended atmosphere” of a WR star. For low mass- +loss rates, the atmosphere is optically thin and the two tempera- +tures align. Although at usually much lower temperatures, this is +similar to what we see for most OB-star winds. With increasing +˙M, we get a more and more extended optically thick layer. Al- +beit leading to a much cooler appearance of the star, this kind of +layer should not be mixed up with the inflated envelope obtained +in various hydrostatic structure models (e.g., Petrovic et al. 2006; +Gräfener et al. 2012; Ro & Matzner 2016). Instead of a subsonic, +but still loosely bound extended layer, our models show super- +sonic (i.e., unbound) layers moving out with hundreds of km s−1. +As illustrated in Sander et al. (2020), the winds often reach more +than 0.5 �∞ before the atmosphere becomes optically thin. In +more recent work (e.g., Poniatowski et al. 2021), this form of +Article number, page 6 of 21 + +A. A. C. Sander et al.: The temperature dependency of Wolf-Rayet-type mass loss +50 +100 +150 +200 +250 +300 +350 +T [kK] +−7 +−6 +−5 +−4 +−3 +log ( ˙M [M⊙ yr−1]) +Teff(τR = 2/3) +Teff(τcrit) +Te(Rcrit) +2.0 Z⊙ +1.5 Z⊙ +1.0 Z⊙ +0.5 Z⊙ +0.2 Z⊙ +0.1 Z⊙ +0.05 Z⊙ +0.02 Z⊙ +Fig. 8. Mass-loss rates as a function of different temperature scales for +the L/M model sequences presented in Sander & Vink (2020). Higher +mass-loss rates correspond to higher L/M-ratios in this plot. +an extended photosphere is termed “dynamical inflation” to dis- +tinguish it from the hydrostatic inflation. Hydrostatic inflation is +not likely to occur if a wind can be launched and maintained, but +it could in situations where the latter is not given. In Sect. 3.2, we +discuss the limits of launching a wind from the hot iron bump. +While a detailed exploration of wind solutions beyond this limit +is not feasible in this study, the numerical results of our “failed” +models indicate a tendency toward larger sonic radii, potentially +indicating some form of hydrostatic inflation. +Finally, we also plot the electron temperature at the criti- +cal (≈ sonic) point as a dotted green curve in Fig. 6. The val- +ues reflect the expected temperature range of the hot iron bump +(around 200 kK), although the particular values are slightly +higher than predicted in structural studies employing OPAL +opacity tables (Grassitelli et al. 2018; Nakauchi & Saio 2018). +The value of Te(Rcrit) appears relatively constant at first sight, +but aside from numerical scatter affecting the results a bit, a sub- +tle trend can be noticed: In the regime of optically thick winds, +there is a tendency toward increasing Te(Rcrit) with higher mass- +loss rates. This is qualitatively in line with the predictions by +Grassitelli et al. (2018), who found that in hydrodynamic stel- +lar structure calculations higher mass-loss rates correspond to +higher temperatures at the sonic point. However, this trend is in- +terrupted when the winds become optically more thin and even- +tually Te(Rcrit) increases mildly with lower ˙M until Teff,crit sur- +passes Te(Rcrit). +All of the temperature trends described for the exemplary +Fig. 6 are observed for the other model sequences as well. For +comparison, we show similar temperature scale plots for the +15 M⊙ sequence (Fig. 7) and the 12.9 M⊙ sequence with surface +hydrogen (Fig. E.3). While there are shifts in the absolute values, +the same general trends are clearly identified. +To study whether our findings are more general or limited +to our new sample, we check the behavior of the different tem- +perature scales also for the whole set of model sequences from +Sander & Vink (2020). The resulting curves are presented in +Fig. 8. Due to the fixed value of T∗(τR,cont = 20) = 141 kK in +Sander & Vink (2020) and the launching of the winds at high +optical depths, the effective temperature referring to the critical +point (i.e., the launch of the wind) hardly varies over the whole +sample. Still, the resulting T2/3-temperatures look very similar to +those obtained in our new models with varying T∗. On the other +hand, Te(Rcrit) varies much more than in any of our new model +R2/3 +1.01 +1.1 +2 +10 +100 +1000 +r/R∗ +arad +acont +apress +athom +AS +-1 +0 +1 +-2 +-1 +0 +1 +2 +3 +log (r/R∗ − 1) +log (a/g) +Fig. 9. Illustration of the radiative acceleration for a model with 10 M⊙ +and T∗ ≈ 115 kK which is not capable of launching a wind from the “hot +iron bump” and thus is dynamically not converged: In the inner part, the +total radiative acceleration (red dashed line) approaches Γrad = 1, but +does not surpass it sufficiently to launch a wind that could be maintained +in the following deceleration region. +sequences. As we also see a shift in the Te(Rcrit)-curves between +different mass sequences in our new work, we can conclude that +Γe – defined by the chemical composition and L/M – plays a ma- +jor role in setting the temperature regime of the sonic point. The +ratio between the flux and the radius – which is mapped in T∗ – +instead only has a minor effect. We do not see the interruption +of the Te(Rcrit)-trend in Fig. 8 that was apparent in Fig. 6 and the +other new model sequences. This is likely due to the different +dimensionality of the sequences in Sander & Vink (2020) (fixed +T∗, variable L/M per sequence) and this work (fixed L/M, vari- +able T∗ per sequence). In general, we can conclude that for stars +further away from the Eddington Limit, the same ˙M can only be +reached by shifting the critical point to lower electron temper- +atures. For the same L/M-ratio, however, we see a much lower +amplitude of changes in Te(Rcrit). In a zeroth-order approxima- +tion, one could state that Te(Rcrit) is constant for a given L/M +and chemical composition. +3.2. WR-type mass loss and its breakdown +The trend of increasing ˙M with lower Teff,crit does not automat- +ically continue beyond the plotted values. The comparison be- +tween the simpler and the more sophisticated treatment in Fig. 7 +already suggests that the effect of deceleration regions has to be +taken into account for computing a more realistic ˙M. In some +situations, such as the one illustrated in Fig. 7, the deceleration +region can become large enough to reduce the wind to subsonic +or even negative velocities, making it impossible to launch a +wind from the deeper layers of the “hot iron bump.” This regime +occurs right next to the (theoretical) maximum of ˙M along the +Teff,crit-axis which is reached when the deceleration region is just +not strong enough to put �(r) below the local sound speed in the +wind. +The situation of a failed wind launch is illustrated in Fig. 9, +where the radiative acceleration barely reaches Γrad = 1 in a +model for 10 M⊙. This example also illustrates that for lower +L/M values, the regime where no wind can be launched from +the hot iron bump gets larger and larger. A hydrogen-rich sur- +Article number, page 7 of 21 + +A&A proofs: manuscript no. paper +Model sequences +20 M⊙, XH = 0.0, Z⊙ +20 M⊙, XH = 0.2, Z⊙ +20 M⊙, XH = 0.2, 0.5 Z⊙ +12.9 M⊙, XH = 0.2, Z⊙ +15 M⊙, XH = 0.0, Z⊙ +20 M⊙, WC, 0.5 Z⊙ +Models with D∞ = 10 +20 M⊙, XH = 0.2, Z⊙ +12.9 M⊙, XH = 0.2, Z⊙ +Models with D∞ = 4 +20 M⊙, XH = 0.2, Z⊙ +AS +-5 +-4 +-3 +-2 +20 +60 +100 +140 +180 +220 +T2/3 [kK] +log( ˙Mt [M⊙ yr−1]) +Fig. 10. Transformed mass-loss rate as a function of T2/3 for our model +sequences +face can compensate this to some degree as it helps to get the +star closer to the Eddington limit. Still, when getting to lower +and lower L/M-ratios the regime of WR winds driven by the hot +iron bump eventually vanishes. In our models with a fixed L-M- +relation this corresponds to a limit in both luminosity and mass. +However, objects of lower masses and luminosity can potentially +launch a wind if they have a considerably higher L/M-ratios than +homogeneous He stars. This might e.g. be the case in WR-type +central stars of planetary nebulae. Gräfener et al. (2017) and Ro +(2019) also pointed out that most of the H-free WN population +in the LMC presents a challenge as these stars should not be able +to launch a wind from the hot iron bump if their masses would +adhere to a typical L-M relation for He-burning stars. However, +this discrepancy is already reduced if we use the Sander & Vink +(2020) models, likely due to the computed flux-weighted opaci- +ties exceeding the OPAL Rosseland opacities assumed as a proxy +for κF in previous studies. The discrepancy could potentially be +reduced even further slightly lower temperatures are considered +as well (cf. Table 3). Nonetheless, the LMC sample remains an +interesting test-bed for detailed comparisons with individual ob- +jects and the limits of radiation-driven winds from the hot iron +bump. +Summarizing the limits of ˙M along the temperature axis, we +see two very different behaviors: Toward cooler temperatures, +we have an abrupt breakdown of the thick wind regime when the +effect of the deceleration region outweighs the initial accelera- +tion by the hot iron bump. This endpoint is reached close to the +highest possible mass-loss rate (for the given stellar parameters) +in this whole wind regime. On the hot temperature end, we in- +stead proceed rather smoothly into the regime of optically thin +winds with lower and lower values of ˙M. This drop along the T- +axis is significantly shallower than the strong breakdown of ˙M +along the L/M-axis we obtained in Sander & Vink (2020). +3.3. Scaling with the transformed mass-loss rate +Since the different calculated model sequences show very sim- +ilar slopes for ˙M(T2/3), we investigate whether there is a com- +mon scaling behind these curves. Given the empirical scaling +relations for WR spectra and our findings from Sander & Vink +(2020), we study the “transformed mass-loss rate” +˙Mt = ˙M +√ +D · +�1000 km/s +�∞ +� �106L⊙ +L +�3/4 +, +(2) +10 +20 +50 +100 +200 +T2/3 [kK] +Empirical results +MW H-free WN-s (Hamann et al. 2006, 2019) +MW H-free WN-w (Hamann et al. 2006, 2019) +MW WCs (Sander et al. 2012, 2019) +LMC H-free WNs (Shenar et al. 2019) +LMC WCs & WOs (Aadland et al. 2022) +SMC WRs (Hainich et al. 2015, Shenar et al. 2016) +Model sequences +20 M⊙, XH = 0.0, Z⊙ +20 M⊙, XH = 0.2, Z⊙ +20 M⊙, XH = 0.2, 0.5 Z⊙ +12.9 M⊙, XH = 0.2, Z⊙ +15 M⊙, XH = 0.0, Z⊙ +20 M⊙, WC, 0.5 Z⊙ +Models with D∞ = 10 +20 M⊙, XH = 0.2, Z⊙ +12.9 M⊙, XH = 0.2, Z⊙ +Models with D∞ = 4 +20 M⊙, XH = 0.2, Z⊙ +AS +4.1 +4.4 +4.7 +5.0 +5.3 +-5 +-4 +-3 +-2 +log( ˙Mt [M⊙ yr−1]) +log(T2/3 [K]) +Fig. 11. Effective temperature at a Rosseland optical depth of 2/3 as a +function of the transformed mass-loss rate ˙Mt for our new calculated +model sequences. The sequences connected by solid lines all employ +D∞ = 50, while those with dashed and dotted curves indicate sequences +using D∞ = 10 and 4, respectively, as indicated in the plot. For compar- +ison, also various empirical results from the literature are depicted by +discrete, gray symbols. +originally introduced by Gräfener & Vink (2013), for our new +model sequences as a function of T2/3. As depicted in Fig. 10, +the resulting curves align extremely well when plotting ˙Mt in- +stead of ˙M. Major offsets are only introduced when assuming +different (maximum) clumping factors D∞. Given our findings +in Sander et al. (2020) and Sander & Vink (2020), the latter is +not much of a surprise. While the clumping does not directly +affect the radiative transfer, the solution of the statistical equa- +tions are solved for a higher density D · ρ (Hamann & Koesterke +1998). This affects the ionization stratification and usually leads +to a larger opacity and thus larger terminal velocity �∞. While +the mass-loss rate ˙M is typically not much affected, the resulting +˙Mt is changed due the increase in �∞ being smaller than the in- +crease in √D∞. For our 20 M⊙ models with XH = 0.2, the typical +increase was about 20% in �∞ when increasing D∞ from 4 to 10 +and about 40% when increasing D∞ from 10 to 50. +To quantify our finding, we flip the axes and show a double- +logarithmic plot in Fig. 11. A clear transition between two +regimes is evident with a “kink” around log ˙Mt ≈ −4.5 that ap- +pears to be independent of D∞. The more dense wind regime +(T2/3 < 130 kK and log ˙Mt > −4.5) can be reasonably well ap- +proximated by a linear fit, yielding +log T2/3 +K += (−0.49 ± 0.01) log +˙Mt +M⊙ yr−1 + (2.91 ± 0.02) +(3) +for the sequences using D∞ = 50. Given the inherent numerical +scatter, in particular in �∞ entering ˙Mt, we can conclude that in +the limit of dense winds T2/3 ∝ ˙M−1/2 +t +. +When comparing the relations with empirically obtained val- +ues of WN (Hamann et al. 2006, 2019; Hainich et al. 2015; +Shenar et al. 2016, 2019) and WC stars (Sander et al. 2012, 2019; +Aadland et al. 2022), it is immediately evident that comparing +T2/3 between empirical and theoretical results yields a much bet- +ter match than the comparison between the empirical T∗ and +our theoretical Teff(τcrit) in Fig. 5 (or the direct T∗-comparison +in Fig. E.1). While the mismatch in Fig. 5 illustrates the “Wolf- +Rayet radius problem” discussed at the beginning of Sect. 3, the +better alignment of the T2/3 values underlines that value of the +empirical analysis, despite the dynamical concerns. In empirical +studies with fixed velocity fields, models are chosen such that +Article number, page 8 of 21 + +A. A. C. Sander et al.: The temperature dependency of Wolf-Rayet-type mass loss +−5.5 +−5.0 +−4.5 +−4.0 +−3.5 +log ( ˙Mt [M⊙ yr−1]) +4.5 +4.6 +4.7 +4.8 +4.9 +5.0 +5.1 +5.2 +log (T2/3 [K]) +2.0 Z⊙ +1.5 Z⊙ +1.0 Z⊙ +0.5 Z⊙ +0.2 Z⊙ +0.1 Z⊙ +0.05 Z⊙ +0.02 Z⊙ +L/M seq. trend +T seq. trend +Fig. 12. Effective temperature at a Rosseland optical depth of 2/3 as +a function of the transformed mass-loss rate ˙Mt for the whole set of +models from Sander & Vink (2020). For log ( ˙Mt [M⊙ yr−1]) < −5.5, +T2/3 is effectively independent of ˙Mt. In Sander & Vink (2020), the +value of T∗ is fixed for all models, but the difference in L/M still yields +a wide range of T2/3 values. The dashed-dotted line represents a linear fit +of the temperature trend for log ( ˙Mt [M⊙ yr−1]) > −4.5 while the dotted +curve represents the fit for the new model sequence illustrated in Fig. 11. +they reproduce the observed spectrum. Although τ2/3 has a sig- +nificant wavelength dependence in WR winds, the effective tem- +perature corresponding to the Rosseland mean value provides +some form of a representative value for the regime that needs to +be met when the light eventually escapes from the star. Our dy- +namically consistent models can generally reproduce these T2/3 +values, but employing more compact radii that better align with +structural predictions. +Despite the generally better match when comparing T2/3, it +is also evident from Fig. 11 that all symbols are either on or left- +ward of the derived curve for D∞ = 50. The most striking dis- +crepancies are obtained for the SMC WN stars. The Aadland +et al. (2022) WC and WO results are very close to our obtained +relation. As they are the only one assuming D∞ = 20, some dis- +crepancies are likely rooted in different clumping assumptions +and treatments. The mismatch of the empirical SMC positions +however, cannot be explained with clumping differences alone +with most of the stars showing empirical ˙Mt values that are about +an order of magnitude lower than our model relations. This could +be due to various effects including considerable differences in +L/M, e.g., due to having significant hydrogen shells and thus not +obeying the assumed L-M-relation in our model sequences, or +too low T2/3 estimates. Investigating these and other possibilities +would add further dimensions to our model sequences and thus +we have to postpone a dedicated analysis of individual targets to +a separate follow-up paper. +When considering the obtained curves in the T2/3- ˙Mt-plane +from the current model sequences, it is so far unclear whether +the slope and even the underlying scaling is universal. In Fig. 12, +we thus plot the same parameters, now using the sequences from +Sander & Vink (2020). A first noticeable difference is the upper +horizontal cutoff at T2/3 ≈ 141 kK, but this is expected due to the +fixed value of T∗ = 141 kK in the Sander & Vink (2020) sample. +The behavior at higher values of ˙Mt looks similar to Fig. 11 at +first – although with considerably more scatter – but an actual +fit of the data reveals a non-negligible difference in the slopes, +Model sequences +20 M⊙, XH = 0.0, Z⊙ +20 M⊙, XH = 0.2, Z⊙ +20 M⊙, XH = 0.2, 0.5 Z⊙ +12.9 M⊙, XH = 0.2, Z⊙ +15 M⊙, XH = 0.0, Z⊙ +20 M⊙, WC, 0.5 Z⊙ +Models with D∞ = 10 +20 M⊙, XH = 0.2, Z⊙ +12.9 M⊙, XH = 0.2, Z⊙ +Models with D∞ = 4 +20 M⊙, XH = 0.2, Z⊙ +AS +-7 +-6 +-5 +-4 +5.0 +5.5 +6.0 +6.5 +log (gcrit [cgs]) +log ( ˙M [M⊙ yr−1]) +Fig. 13. Mass-loss rate ˙M as a function of the gravitational acceleration +at the critical radius gcrit = GMR−2 +crit. Thin, dotted, gray lines indicate +curves with ˙M ∝ g−3/2 +crit . +yielding +log T2/3 +K += (−0.667 ± 0.009) log +˙Mt +M⊙ yr−1 + (2.21 ± 0.03) +(4) +for the Sander & Vink (2020) sample. (For comparison, the de- +rived trend for the new sequences is shown as well in Fig. 12.) +We discuss possible origins later in Sect. D. +3.4. Quantitative mass-loss radius-dependence +The similarity of the curves in Fig. 5 and the simplicity of the +slopes, indicates a common dependence between log ˙M and +log Teff,crit for our sequences with fixed L/M. Performing a linear +fit, we obtain a relation in the form of +log( ˙M [M⊙ yr−1]) = −6 · log(Teff,crit [K]) + offset +(5) +with the detailed fit coefficients being presented in appendix +Sect. A and Table A.1. The factor −6 in Eq. (5) implies that the +obtained temperature dependence essentially reflects the radius +change of the stellar models, since T 4 +eff,crit ∝ R−2 +crit and +˙M ∝ R3 +crit +(6) +for models with L = const. (as in our model sequences). Our +corresponding plot (Fig. A.1) further shows that deviations from +the purely geometrical trend occur when we reach the limit of +radiative driving discussed in Sect. 3.2 (and listed explicitly for +each sequence in Table A.1), as e.g. visible at the upper end of +the WC sequence (cf. Fig. A.1). +From a dynamical perspective, the obtained Rcrit-trend for ˙M +can be understood as a dependence on the gravitational acceler- +ation on the critical point, where the wind is launched. With the +straight-forward definition of +gcrit = g(Rcrit) = GM +R2 +crit +(7) +we can rewrite Eq. (6) as +˙M ∝ g−3/2 +crit +(8) +since M is a constant among each of the model sequences. Trend +curves reflecting Eq. (8) are displayed in Fig. 13 together with +Article number, page 9 of 21 + +A&A proofs: manuscript no. paper +the curves from our model sequences. Generally, a decreasing +trend of ˙M with log gcrit is not surprising as an increased grav- +itational force needs to be overcome. Given the content mass +M along the model sequences, the change in gcrit expected from +Eq. (8) is purely geometrical, that is only from the change in Rcrit. +The model sequences align well with Eq. (8), but there is a no- +table flattening for the highest mass-loss rate, that is in the case +of more dense winds. We cannot rule out that a numerical ef- +fect is playing a role here as these high- ˙M models often operate +on the limits of what the code is capable of. Nonetheless, given +that the bending occurs in all sequences, a physical origin seems +more likely and we continue our efforts on this assumptions. For +some sequences, there are also notable deviations from Eq. (8) +at the lower ˙M-end. From the current set of calculations, the ap- +parent kink in the curves approximately coincides with Te(Rrcrit) +surpassing Teff,crit, meaning that the electron temperature at the +critical point is higher than the effective temperature at this point. +However, the low number of models where this trend is clearly +observed and the need to include higher ionization stages in these +models, which can cause an additional offset in the numerical +solutions if not done early enough in the sequence, currently re- +frain us from concluding whether there is a clear “kink” or a +more gradual change that might potentially be emphasized by a +switch in the numerical setup. In any case, the model solutions +obtained in this thinner wind regime are characterized by (elec- +tron) temperature stratification that remain very high, e.g. larger +than 50 kK, until infinity. Their leading acceleration is provided +by Fe M-shell ions, which are populated throughout the wind, +qualitatively similar to the example shown in Fig. 16 of Sander +et al. (2020). When considering the transformed mass-loss rate +˙Mt instead of ˙M, the scaling of the velocity with gcrit has to be +considered as well, which we do in the following section. +4. Terminal velocity trends +With the intrinsic solution of the hydrodynamic equation of mo- +tion, our models automatically predict terminal wind velocities +together with ˙M. While already entering the transformed mass- +loss rates, we now take a look at the explicit results for �∞ as +a function of T2/3 in Fig. 14. Although our sequences are not at +all adjusted to match any particular observations, we also plot +empirical results for WN stars obtained with PoWR for com- +parison. It is clear that our models match the general regime of +the observed sample, but a closer inspection also shows caveats, +for example with hydrogen-free WN stars showing values above +the 20 M⊙ H-free sequence. Various possibilities could explain +this (e.g., higher L/M and/or higher clumping), but a thorough +investigation is beyond the scope of this paper. +4.1. Impact of clumping +In contrast to ˙M, the values for �∞ tend to scatter a bit more due +to being evaluated at the outer boundary of the models. The ter- +minal velocity further strongly depends on the included opacity, +so including all ions contributing to the acceleration is necessary +in order to avoid underestimating �∞. As apparent from Fig. 14, +�∞ also reacts on the choice of the clumping factor D∞. With +a depth-dependent onset of the clumping, the response of the +mass-loss rate to a change of D∞ is usually small as the result- +ing differences in D(r) are small in the subsonic layers (see also +Fig. 1 in Sander et al. 2020, and appendix Sect. C of this work). +In the supersonic layers, however, any differences in D∞ affect +the bulk of the opacities being considered in the hydrodynamic +equation of motion. Consequently, the obtained values for �∞ +Empirical results +MW H-free WN-s +MW H-free WN-w +LMC H-free WNs +SMC WRs +Model sequences +20 M⊙, XH = 0.0, Z⊙ +20 M⊙, XH = 0.2, Z⊙ +20 M⊙, XH = 0.2, 0.5 Z⊙ +12.9 M⊙, XH = 0.2, Z⊙ +15 M⊙, XH = 0.0, Z⊙ +20 M⊙, WC, 0.5 Z⊙ +Models with D∞ = 10 +20 M⊙, XH = 0.2, Z⊙ +12.9 M⊙, XH = 0.2, Z⊙ +AS +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +20 +60 +100 +140 +180 +220 +T2/3 [kK] +v∞ [km s−1] +Hawcroft et al. (ULLYSES) +Vink & Sander (2021) +Fig. 14. Terminal velocity as a function of T2/3 for the different model +sets. For comparison, also the derived trend for OB-star winds in +the Milky Way (dashed line) and the LMC (dashed-dotted line) from +Hawcroft et al. (in prep.) are shown. +are notably higher for higher D∞, typically on the order of 20% +when calculating a model for the same L, M, T∗, and chemical +composition with D∞ = 50 instead of 10. In particular cases, the +effects can be much larger, e.g. when a model has a deceleration +regime in the case of a lower D∞, while there is no such regime +for higher D∞. Significant changes of the wind density regime +due to a switch of D∞ can then lead to a stronger change in the +derived ˙M. Moreover, the assumption of little to no clumping +in the deeper layers would become invalid in case of a porous, +optically thick medium, which can result from a subsonic, super- +Eddington situation (e.g., Shaviv 1998, 2000). +4.2. Scaling with T2/3 +Beside the differences due to clumping, Fig. 14 demonstrates +that any significant change of the chemical composition usu- +ally affects the derived �∞-values. In the figure, we see higher +terminal velocities for the 20 M⊙ model sequence with surface +hydrogen (XH = 0.2) compared to the corresponding hydrogen- +free sequence. This result might be counter-intuitive at first as +hydrogen does not provide significant line opacity that could be +used to increase �∞. However, the additional hydrogen is able to +boost the mass-loss rate of the star as the hydrogen atoms in the +atmosphere provide a higher budget of free electrons compared +to a hydrogen-free atmosphere1. With more acceleration avail- +able already in the deeper layers, the critical point of the wind +moves inward to higher optical depths. Line opacities which +were subsonic in the hydrogen-free case can now be used to fur- +ther boost the terminal wind speed. Due to the higher mass loss +of the hydrogen-containing model, the value of T2/3 decreases +when comparing models with the same T∗. Interestingly, as we +saw in Fig. E.1, the value of ˙Mt remains the same when com- +paring against T2/3. In a follow-up study, we will test whether +this behavior is universal when considering surface hydrogen or +whether the chosen fraction of XH = 0.2 coincidentally balances +out other effects for a 20 M⊙ He-burning star as we e.g. saw with +the metallicity reduction being balanced by the surface hydrogen +when considering only ˙M in Fig. 5. +1 A small hydrogen-layer on the surface has also the structural conse- +quence of an increased stellar radius, which would again affect the wind +parameters. Here, we discuss only the immediate atmospheric conse- +quences for a fixed set of stellar parameters. +Article number, page 10 of 21 + +A. A. C. Sander et al.: The temperature dependency of Wolf-Rayet-type mass loss +To get some insights on the general scaling of WR-type +winds with effective temperature (here: T2/3), we also compare +our sequences to the trends for OB-type stars. In Fig. 14, we plot +the trends obtained for the ULLYSES OB stars for the SMC and +a Galactic comparison sample by Hawcroft et al. (in prep.) as +well as the predictions from Vink & Sander (2021). We see that +generally the terminal velocity increases much steeper with the +temperature in OB-type winds than in WR-type winds. Only at +higher temperatures (T2/3 > 130 kK), when the winds become +more optically thin as the mass-loss rates decrease, the steepness +of the �∞(T2/3)-curves increases to a value more comparable to +those obtained for OB-star winds. +4.3. Critical-point dependencies +In addition to studying the behavior of �∞ as a function of the +observable quantity T2/3, we further investigate the behavior of +�∞ as a function of T∗ or the physically more meaningful Teff,crit. +As noted above, the values of �∞ tend to scatter a bit more, but +if we restrict the linear fitting to the optically thick wind regime, +we find +log +� +�∞ [km s−1] +� += 1.2 log (Teff(τcrit) [K]) + offset. +(9) +Given that the number of models in the optically thick regime +is restricted and not all sequences reach far enough to see the +flattening of the trend, this coefficient has to be considered as +rather uncertain. Nonetheless, the corresponding scaling of �∞ ∝ +R−0.6 +crit can also be obtained from directly fitting �∞(Rcrit) in the +optically thick limit. Using the critical radius to define the escape +velocity +�esc = +� +2GM +Rcrit +(10) +we obtain +�∞ ∝ �1.2 +esc. +(11) +This relation also holds for the effective escape velocity +�esc = +� +2GM +Rcrit +(1 − Γe) +(12) +since all of the model sequences have a constant L/M and Γe +is approximately constant. The latter is consequence of the un- +changed free electron budget below Rcrit in the considered tem- +perature range. Thus, in the optically thick wind regime we have +a slight difference with �∞ ∝ �1.2 +esc,eff along the T∗-dimension com- +pared to the well-known �∞ ∝ �esc,eff in the well-known CAK +theory (named after Castor, Abbott, & Klein 1975). For the se- +quences along the L/M-dimension from Sander & Vink (2020), +we instead obtain a negative trend of log �∞ ≈ −4.6 log �esc,eff + +const. in the optically thick regime with a flattening of the trend +for the highest mass-loss rates. In both cases, the scaling of �∞ +with �esc,eff remains complicated with no straight-forward predic- +tion as offsets remain in all scalings. This is in sharp contrast to +the classical (m)CAK result, where �∞ follows as an offset-free +value from �esc. +For lower mass-loss rates (log ˙Mt < −4.5), corresponding +usually to Teff,crit > 150 kK, we reach the regime where winds are +mostly optically thin. Above, we could show that when reaching +this regime, there seems to be an alignment of �∞(T2/3), with the +slopes known from OB-type winds. With considerable scatter in +the exponent of up to ±0.5, we find +�∞ ∝ T 4 +eff,crit, +(13) +Model sequences +20 M⊙, XH = 0.0, Z⊙ +20 M⊙, XH = 0.2, Z⊙ +20 M⊙, XH = 0.2, 0.5 Z⊙ +12.9 M⊙, XH = 0.2, Z⊙ +15 M⊙, XH = 0.0, Z⊙ +20 M⊙, WC, 0.5 Z⊙ +Models with D∞ = 10 +20 M⊙, XH = 0.2, Z⊙ +12.9 M⊙, XH = 0.2, Z⊙ +Models with D∞ = 4 +20 M⊙, XH = 0.2, Z⊙ +AS +-6 +-5 +-4 +-3 +5.0 +5.5 +6.0 +6.5 +log (gcrit [cgs]) +log ( ˙Mt [M⊙ yr−1]) +Fig. 15. Transformed mass-loss rate ˙Mt as a function of the gravita- +tional acceleration at the critical radius gcrit = GMR−2 +crit. To reflect the +expected trends from the Rcrit-fits, thin gray lines are plotted in the +background. The dotted, gray lines indicate ˙M ∝ g−2.5 +crit (optically thin +regime), while the dashed, gray lines correspond to ˙M ∝ g−1.8 +crit (optically +thick regime). +corresponding to �∞ ∝ R−2 +crit or �∞ ∝ �4 +esc,eff, that is a steeper re- +lation, contrary to the expected flattening of the slope. However, +there is growing evidence from both observations (e.g., Garcia +et al. 2014) as well as theoretical CMF-based and Monte Carlo +calculations (e.g., Björklund et al. 2021; Vink & Sander 2021) +that even in the typical OB-type regime the scaling of �∞ with +�esc,eff is likely more complicated. +4.4. Influence on ˙Mt +The scaling of �∞ with Rcrit introduces an additional dependency +when considering the transformed mass-loss rate ˙Mt as a func- +tion of Rcrit or gcrit. +From Eqs. (5) and (6), we know that ˙M ∝ T −6 +eff,crit or ˙M ∝ R3 +crit. +From the definition of ˙Mt (Eq. 2) we get +log ˙Mt(Rcrit) = log ˙M(Rcrit) − log �∞(Rcrit) + offset. +(14) +With the different trends derived for the optically thick and thin +limit in Sect. 4.3, we obtain ˙Mt ∝ R3.6 +crit and ˙Mt ∝ R5 +crit respec- +tively. Using gcrit as defined in Eq. (7) and Eq. (8), this yields +log ˙Mt = 1.8 log gcrit + offset +(15) +for the optically thick limit and +log ˙Mt = 2.5 log gcrit + offset +(16) +in the optically thin limit. These trends are depicted as sets of +gray lines in Fig. 15, where the curves from the model sequences +are shown as well. In contrast to the ˙M(gcrit)-behavior discussed +in Sect. 3.4, the representation of the slope in the optically thin +regime is less precise. In the optically thick limit, some curves +align well, but others appear to be slightly steeper or shallower. +Hence, the overall results for ˙Mt(gcrit) should be considered less +robust than the ˙M(gcrit)-trends. Interestingly, we do not see the +“kink” or clear bending for some sequences at lowest (trans- +formed) mass-loss rates that we see in Fig. 13 for ˙M(gcrit). While +it is hard to draw strong conclusions, there at least appears to +be one continuous slope for ˙Mt(gcrit) in the thinner wind regime, +regardless of whether the critical point is located at temperatures +Article number, page 11 of 21 + +A&A proofs: manuscript no. paper +below or above Teff,crit. Since we find a change for ˙M alone, this +would imply that �∞ outweighs this effect. While the inspection +of the corresponding sequences in Fig. 14 is only indicative here, +indeed the �∞-curves of these sequences bend again toward shal- +lower slopes then plotting them as functions of Teff,crit. +5. Potential consequences for stellar evolution +Our study presents the very first sequences of hydrodynamically +consistent atmosphere models in the cWR regime, where we +vary the input parameter T∗ – corresponding roughly to Teff,crit +for most models. WR mass-loss recipes commonly do not incor- +porate any temperature/radius dependency, which can be seen as +a consequence of the optically dense winds of WR stars. When +reproducing their spectra with prescribed velocity fields, there is +a degeneracy of solutions making it impossible to find a unique +value of T∗ for more dense winds (e.g., Hillier 1991; Hamann & +Gräfener 2004; Lefever et al. 2022). +From an evolutionary standpoint, one could justify the omis- +sion of a T∗-dependence arguing that hydrogen-free WN stars – +and to some extend also WC stars – may form a 1D sequence as +they represent He-burning stars that do not contain any further +shell structure which could skew the relation between the lumi- +nosity and mass. In reality, effects such as inflation, convection, +or rotation augment the physical conditions of wind launching +and mass loss, especially when considering their multidimen- +sional nature. However, in the currently typical 1D spherical ap- +proach ignoring such issues, no further parameter would be nec- +essary if the He star evolution could be perfectly mapped to one +of the fundamental stellar parameters. For He stars above 10 M⊙, +the intrinsic curvature of the HeZAMS indeed gets relatively +small (e.g., Langer 1989) and the obtained tracks of WR evo- +lution in different codes yield very similar temperatures around +log(T [K]) = 5.12, regardless whether these have been calcu- +lated from pure He stars or including all prior evolution from +the ZAMS (e.g., Georgy et al. 2012; Limongi & Chieffi 2018; +Higgins et al. 2021). +5.1. Mass-loss comparison for a representative model +In the models from Sander & Vink (2020), we thus ignored the +width of ≈ 0.1 dex in log T∗ and fixed T∗ in order to keep the +total amount of models manageable. In this work, we now cal- +culated a number of model sequences where we vary T∗ in order +to investigate the effect of a wider range of T∗, which probes not +only the curvature of the He ZAMS, but also gives a glimpse +of how ˙M might be affected for stars which are not yet or no +longer (exactly) on the He ZAMS. We find that despite the nar- +row range in temperature, the effect on ˙M is quite noticeable. +For our 20 M⊙ model sequence (at Z⊙) even a narrow range of +only 0.05 dex (i.e., T∗ = 125 . . . 140) results in a factor of two in +˙M. Whether such a significant correction is really necessary de- +pends on the difference between the most realistic choice of T∗ +and the fixed value (141 kK) in Sander & Vink (2020). Combin- +ing the structural constraints by Grassitelli et al. (2018) with our +model sequence, we find an ideal value of Teff,crit ≈ T∗ ≈ 130 kK +for a 20 M⊙ at Z⊙ model without any hydrogen. +In Table 3, we provide a comparison of the resulting mass- +loss rates for a 20 M⊙ star. Beside the values employing the new +2 In this discussion, we do not consider structure models that show hy- +drostatic envelope inflation for more massive He stars (see, e.g., Fig. 19 +in Köhler et al. 2015) as Grassitelli et al. (2018) demonstrated that such +an inflation likely does not occur if a strong wind can be launched. +Table 3. Comparison of mass-loss rates obtained with different methods +for a hydrogen-free WN star with log L/L⊙ = 5.7 and 20 M⊙ +Paper +log( ˙M [M⊙ yr−1]) +Z⊙ +0.5 Z⊙ +Gräfener et al. (2017), s.-analytic(a) +−4.72 +no sol. +Gräfener et al. (2017), num.(b) +−4.65 +no sol. +Sander & Vink (2020) (D∞ = 50) +−4.61 +−4.75 +Sander & Vink (2020) (D∞ = 10) +−4.64 +−4.83(c) +this work, T∗ = 130 kK(d) (D∞ = 50) +−4.40 +−4.56 +this work, T∗ = 130 kK(d) (D∞ = 10) +−4.42 +−4.83(e) +Nugis & Lamers (2000) recipe +−4.52 +−4.66 +Hamann95+(f) recipe +−4.40 +−4.65 +Yoon (2017) recipe ( fWR = 1) +−4.59 +−4.77 +Yoon (2017) recipe ( fWR = 1.6) +−4.39 +−4.57 +Notes. The ˙M determinations by Gräfener et al. (2017) employ the Prad- +Pgas-plane with the sonic point conditions using (a) their Eq. (27) and +assuming �∞ = 1800 km s−1 or (b) a numerically integrated dPrad/dr . +(c) New calculation, but with T∗ = 141 kK as in Sander & Vink (2020). +(d) Choice of T∗ based on matched Teff(τcrit) with Grassitelli et al. (2018) +(e) Unstable solution close to driving breakdown, see Sect. 3.2 +(f) Mass-loss rates from Hamann et al. (1995) divided by a factor 10 +and scaled with the Z-dependence from Vink & de Koter (2005), as +suggested by Yoon et al. (2006). +estimate of T∗ and the solutions for T∗ = 141 kK from Sander & +Vink (2020), we also list the resulting ˙M-values from Gräfener +et al. (2017) and commonly used (semi-)empirical recipes. For +Z⊙ we find a difference of ∼0.2 dex in +˙M, unaffected by the +choice of D∞. Using the values of Table 3 as an average mass- +loss rate during the typical He burning lifetime (300 kyr), this +corresponds to a difference between 12.5 M⊙ and 8.1 M⊙ at the +end of core He-burning. This calculation is of course only a +rough estimate and does not take any change of the stellar pa- +rameters or surface abundances into account. Nonetheless, the +value using the ˙M from Sander & Vink (2020) is close to what +we obtain with actual stellar evolution calculations in Higgins +et al. (2021). +At 0.5 Z⊙, a value roughly corresponding to the LMC, the +0.2 dex shift holds as well when adopting D∞ = 50. For the +hydrogen-free 130 kK model at 0.5 Z⊙ with D∞ = 10, however, +we only find a solution if we relax the stability criterion on ˙M +between consecutive updates that we otherwise enforce. For the +141 kK model, we already see a notable difference in ˙M when +reducing from D∞ = 50 down to 10. The reason is that we are +already close to the regime where we can no longer obtain a +wind solution driven by the hot iron bump (see Sect. 3.2). For +the 130 kK we have reached already a meta-stable situation with +respect to the solution stability. Thus, the obtained value of ˙M for +D∞ = 10 is much lower than expected. The matching of the ab- +solute values for 130 kK and 141 kK is a pure coincidence with +higher, also meta-stable solutions up to ≈ −4.7 for 130 kK being +possible as well. +5.2. Structural limits and the role of hydrogen +The presence of hydrogen at the surface can considerably change +the limits of the wind onset derived above. In contrast to our +hydrogen-free results shown in Table 3 and depicted in Fig. 6, +our model sequence with XH = 0.2 and 0.5 Z⊙ extends to much +cooler temperatures (Teff,crit < 100 kK) as the additional accel- +eration from free electrons helps to compensate the effect of the +Article number, page 12 of 21 + +A. A. C. Sander et al.: The temperature dependency of Wolf-Rayet-type mass loss +deceleration regions. The choice of D∞ has some impact on the +results, but in both cases the effect of surface hydrogen as such +is much larger. +While we do not aim at a detailed comparison with obser- +vations in this work – which would require new analyses with +dynamically-consistent models – it is striking that all hydrogen- +free WN stars in the LMC are of the subtype WN4 or earlier +(Hainich et al. 2014; Shenar et al. 2019). Moreover, all WC +stars in the LMC show early subtypes as well. While one has to +be careful drawing absolute conclusions, our temperature study +now indicates that beside the metallicity limiting the lower lu- +minosity of the observed WN population, the observed restric- +tion of the subtype regime might be a direct consequence of the +inability to launch WR-type winds below a certain (sonic point) +temperature. From the perspective of fixed stellar parameters, the +lower mass-loss rate reached at a lower metallicity corresponds +to a shift to earlier subtypes (cf. Fig. 4), thereby confirming the +suggestion by Crowther et al. (2002). +Coming from a different angle, but addressing essentially +the same problem, Grassitelli et al. (2018) and Ro (2019) used +hydrodynamic stellar structure models and semi-analytic ap- +proaches to predict the existence of a “minimum mass-loss rate” +for launching a stellar wind from the hot iron bump. For values of +˙M below this limit, extended low-density regions were predicted. +In this work, we do not aim to obtain the latter type of solutions, +but the calculations failing to launch a wind show a tendency +toward trying to launch a wind further out with a (much) lower +˙M. We can thus qualitatively confirm the structural predictions +by Grassitelli et al. (2018) and Ro (2019), assuming that we are +bound to a choice of T∗ following the – ideally hydrodynamical +– structure calculations for the HeZAMS. This underlines once +more that unifying structural and atmosphere models remains a +challenge that requires a new generation of both atmosphere and +structure models. +5.3. An approximated handling of the temperature shift +In light of the structural considerations above, it appears likely +that any future description of WR-type mass loss needs a tem- +perature or radius-dependency. Simpler treatments might be rea- +sonable in a time-averaged situation, but cannot predict a real- +istic mass loss for individual points along an evolutionary track. +Hence, a detailed update of the Sander & Vink (2020) formula +will eventually be necessary, but the current amount of models +does not allow a wide-space parameter investigation. The appli- +cability to lower masses also turns out to be nontrivial: One the +one hand, the curvature of the HeZAMS toward cooler temper- +atures should soften the sharp drop obtained in Sander & Vink +(2020) of ˙M toward lower He star masses. On the other hand, +we reach lower limits of Teff,crit for driving winds by the hot iron +bump (cf. Sect. 3.2). In our 12.9 M⊙ sequence with XH = 0.2, +this limit is at ≈ 97 kK. Given that this limit seems to increase to +slightly higher temperatures for lower L/M values, it appears un- +likely that one can find any solution for a wind driven by the hot +iron bump for stars with M ≤ 10 M⊙ fulfilling the L-M relation +from Gräfener et al. (2011). +While a full coverage of the driving limit at cooler tempera- +tures will require its own tailored study, we can use Eq. (5) from +Sect. 3.4 to derive a decent temperature description up to dis- +continuity in ˙M. Since Eq. (5) seems to be valid across both the +optically thick and thin regime, we can approximate ˙M for WN +winds driven by the hot iron bump via +log +� +˙M +M⊙ yr−1 +� += log +� ˙MSV2020 +M⊙ yr−1 +� ++ 3 log +� +Rcrit +Rcrit,T141 +� +(17) +with ˙MSV2020 denoting the mass-loss rate from Sander & Vink +(2020) and Rcrit,T141 = Rcrit(T∗ = 141 kK) being the critical ra- +dius (in R⊙) of their corresponding model (e.g., 1.217 R⊙ for the +20 M⊙ He star without hydrogen). Although Rcrit,T141 could be +obtained from L/M or Γe via a nonlinear fit of the Sander & Vink +(2020) data, it is much more convenient to reformulate Eq. (17) +in terms of the effective temperature at the critical (≈ sonic) point +Teff,crit, yielding +log +� +˙M +M⊙ yr−1 +� += log +� ˙MSV2020 +M⊙ yr−1 +� +− 6 log +� Teff,crit +141 kK +� +. +(18) +Apart from small deviations for the highest mass-loss rates +( ˙M ≫ 10−4 M⊙ yr−1), the fixed value of 141 kK accurately repre- +sents the value of Teff,crit in Sander & Vink (2020), as illustrated +previously in Fig. 8. +This adjusted +˙M-recipe requires the knowledge of either +Teff,crit or Rcrit. As we did include only a small microturbulent ve- +locity in our modeling efforts (30 km s−1), the quantities can be +replaced by the sonic point values without introducing a consid- +erable error. Still, the accurate use of Eq. (17) and (18) requires +models with a meaningful sonic point in a hydrodynamical sense +to prevent reintroducing any further radius/temperature discrep- +ancies. Stellar atmosphere analyses typically employ predefined +velocity fields (usually β-laws) and thus do not have a sonic point +that is hydrodynamically consistent. Purely hydrostatic stellar +structure calculations are problematic as well as they do not yield +a sonic point by construction. This underlines that in order to ob- +tain a really insight- and meaningful comparison between theory +and observation for optically thick winds, a new generation of +both atmosphere and stellar structure models will be necessary. +The results obtained in our study could be helpful to even- +tually obtain realistic predictions for the effective temperatures +(T2/3) of WR stars in stellar evolution models. A route toward +such a recipe based on our findings is given in appendix Sect. D. +6. Transparency to He II ionizing photons +Despite having intrinsically quite hot temperatures, classical WR +stars do not necessarily emit a significant number of ionizing +photons beyond the He ii ionization edge, that is below 227 Å +or above 54 eV. As first described in Schmutz et al. (1992), the +transparency of the wind for photons with energies above 54 eV +depends on the mass-loss rate, and thus the density of the wind. +In more dense winds, He iii recombines to He ii, making the at- +mosphere opaque to He ii ionizing photons out to very large radii. +The presence of line blanketing further affects the absolute ion- +izing fluxes significantly (cf. Smith et al. 2002). Beside usually +leading to a reduction of the He i ionizing flux, it can also affect +the He ii ionizing flux transition by a few orders of magnitude +as we will see in our model calculations. To cover the region +where the (continuum) optical depth drops below unity in this +wavelength region, we extended the outer boundary radius Rmax +of our atmosphere models to extremely large values, often up to +100 000 R∗. (Typical atmosphere models for spectral fitting re- +quire only Rmax = 100 . . . 1000 R∗.) +In Fig. 16, we plot the rate of ionizing photons per second +QHe ii as a function of the transformed mass-loss rate ˙Mt. The ab- +solute numbers in the regime with low QHe ii are more uncertain +Article number, page 13 of 21 + +A&A proofs: manuscript no. paper +Model sequences +20 M⊙, XH = 0.0, Z⊙ +20 M⊙, XH = 0.2, Z⊙ +20 M⊙, XH = 0.2, 0.5 Z⊙ +12.9 M⊙, XH = 0.2, Z⊙ +15 M⊙, XH = 0.0, Z⊙ +20 M⊙, WC, 0.5 Z⊙ +Models with D∞ = 10 +20 M⊙, XH = 0.2, Z⊙ +12.9 M⊙, XH = 0.2, Z⊙ +Models with D∞ = 4 +20 M⊙, XH = 0.2, Z⊙ +AS +38 +41 +44 +47 +50 +-5 +-4 +-3 +log( ˙Mt [M⊙ yr−1]) +log(QHeii [s−1]) +Fig. 16. Number of helium ionizing photons per second QHe ii as a func- +tion of the transformed mass-loss rate ˙Mt for the new model sequences +calculated in this work. +−7.0 +−6.5 +−6.0 +−5.5 +−5.0 +−4.5 +−4.0 +−3.5 +log ( ˙Mt [M⊙ yr−1]) +38 +40 +42 +44 +46 +48 +50 +log QHe ii +2.0 Z⊙ +1.5 Z⊙ +1.0 Z⊙ +0.5 Z⊙ +0.2 Z⊙ +0.1 Z⊙ +0.05 Z⊙ +0.02 Z⊙ +Fig. 17. Number of helium ionizing photons per second QHe ii as a func- +tion of the transformed mass-loss rate ˙Mt for the model sequences com- +puted in Sander & Vink (2020). +as they depend strongly on the precise boundary treatment. If +the wind becomes transparent around and below 227 Å only very +close to the model boundary or even remains optically thick at +Rmax, the value for QHe ii can be underestimated, but should never +exceed 1041 s−1. Given that this is many orders of magnitude be- +low the actually strong QHe ii emitters with rates > 1047 s−1, the +values of the shaded regime in Fig. 16 should have no practical +consequences. +Displaying the He ii ionizing flux as a function of +˙Mt in +Fig. 16 confirms that similar to other quantities, the switch +in transparency is caused by the lower wind density. In +fact, there seems to be a critical lower boundary around +log( ˙Mt [M⊙ yr−1]) = −4.6 to −4.4 where all model sequences +switch abruptly. To study whether this transition value might +be more universal, we created the same plot for the model se- +quences from Sander & Vink (2020). Their model set, shown in +Fig. 17, clearly hints at a Z-dependency for the transition, which +we do only sparsely map in our new model set. In fact, our new +sequences are quite complementary to the datasets from Sander +& Vink (2020), indicating that the transition does not only de- +pend on Z as a total value, but likely on the detailed composi- +tion and – notably especially at the lower end of the transition +in Fig. 16 – on the choice of the clumping factor. Nonetheless, +we can conclude that all stars in our large model sample with +log( ˙Mt [M⊙ yr−1]) < −4.6 are strong emitters of He ii ionizing +flux. Thus, we propose to take this value as an upper limit of +whether to consider WR stars as notable contributors to the He ii +ionizing photon budget. +7. Summary and conclusions +In this work, we presented an exploratory study for the +temperature-dependency of radiation-driven winds launched by +the so-called hot iron opacity bump. For the first time, we cal- +culated temperature-dependent sequences of hydrodynamically +consistent stellar atmosphere models in the cWR regime. To +achieve our results, we had to allow for nonmonotonic velocity +field solutions when solving the hydrodynamic equation of mo- +tion. In order to perform the necessary radiative transfer in the +comoving frame, we afterwards interpolated the obtained veloc- +ity fields such that the main wind properties ( ˙M, �∞) as well as +the characteristics in the outer wind were maintained. We draw +the following conclusions: +– The mass-loss rates ˙M depend significantly on the critical +radius Rcrit and thus also on the assumed model temperature +setting Teff(Rcrit). For model sequences with constant lumi- +nosity L and stellar mass M, we obtain ˙M ∝ R3 +crit over a +wide range with moderate deviations from this purely geo- +metrical effect occurring at the lower and upper end of our +sequences. This finding can also be expressed in the form +of ˙M ∝ g−3/2 +crit , reflecting that larger radii for the critical point +imply a lower gravitational force. Our findings underline that +WR-type mass-loss depends on multiple parameters and the +2D description from Sander & Vink (2020) needs to be ex- +tended further to describe all relevant effects. +– Except +for +very +dense +winds +– +corresponding +to +log( ˙Mt [M⊙ yr−1]) +≈ +−3.0 and above – the effective +temperature at the critical point Teff(τcrit) is close to the +effective temperature at a Rosseland continuum optical +depth of τR,c = 20. For WN-type models τR,c = 20 typically +corresponds to τThom ≈ 17, albeit with considerable scatter +along the model sequences. +– We find a characteristic value of log( ˙Mt [M⊙ yr−1]) ≈ −4.5 +for the transition between the optically thin and thick regime. +While there is some scatter between different model se- +quences, this characteristic value of ˙Mt (plus some error mar- +gin) provides a very convenient tool to distinguish between +the regimes as ˙Mt can also be determined with empirical +models. Known WC stars show values well above this (e.g., +Gräfener & Vink 2013; Sander et al. 2019) while WO stars +might be found on both sides of the transition. Whether the +characteristic value of ˙Mt also holds for winds that might not +be driven by the hot iron bump is currently unclear. We cal- +culated the transformed mass-loss rates for stars at the spec- +tral transition from Of to WNh, which likely happens at a +cooler temperature regime than studied in this work3. Their +corresponding transformed mass-loss rates ˙Mt,trans appear to +be below −4.5, e.g. at log( ˙Mt,trans [M⊙ yr−1]) ≈ −5.0 in the +Arches cluster (Martins et al. 2008; Vink & Gräfener 2012) +3 The transformed mass-loss rate ˙Mt as such should not be confused +with the transition mass-loss rate ˙Mtrans from Vink & Gräfener (2012). +Nonetheless, if the other necessary parameters are known, one can es- +timate the corresponding transformed mass-loss rates for stars defining +the transition mass-loss rate, denoted as ˙Mt,trans +Article number, page 14 of 21 + +A. A. C. Sander et al.: The temperature dependency of Wolf-Rayet-type mass loss +and even lower in R136 (Bestenlehner 2020; Bestenlehner +et al. 2020). +– The choice of the maximum clumping factor D∞ does not +affect our derived ˙M(T2/3) trends, but leads to an additional +shift in the obtained relations with higher clumping factors +corresponding to higher ˙M for the same T2/3. In contrast to +all shifts introduced by varying fundamental stellar parame- +ters or abundances, the shift due to D∞ does not vanish when +considering ˙Mt(T2/3) instead of ˙M(T2/3). +– In the limit of optically thick winds, we obtain a linear re- +lation between log T2/3 and log ˙Mt, independent of chem- +ical composition (but for a fixed clumping factor). Com- +bined with the also Z-independent result from Sander & Vink +(2020) that log ˙Mt ∝ log(L/M), this could provide an easy- +to-use prediction for WR effective temperatures in stellar +structure and evolution models. +– Classical +WR +stars +and +non-WR +helium +stars +with +log( ˙Mt [M⊙ yr−1]) < −4.6 are strong emitters of He ii ion- +izing flux (with QHe ii > 1048 s−1). Helium stars with stronger +winds are (mostly) opaque to radiation above 54 eV and thus +should not be considered as sources of hard ionizing radia- +tion. +– Albeit being limited in comparability to particular observed +targets, our findings indicate that high clumping factors (D ≈ +50) might be necessary to reproduce the observed combina- +tions of ˙M and �∞. This is in sharp contrast to the first results +obtained from 3D wind modeling by Moens et al. (2022) ar- +guing for much lower clumping factors of D ≈ 2. At present, +the reason for this discrepancy is unclear. Various solutions +are possible, e.g., missing opacities in our wind models – +where the presently assumed high clumping would act as a +“fudge factor” to make up for that. Alternatively, the sharp +contrast in the clumping factor might simply be the result of +a mismatch between the considered regimes. Currently, the +3D models from Moens et al. (2022) probe only the wind +onset where even in our 1D models we assume D(r) ≪ D∞ +with D(τcrit) typically ranging between 1.5 and 4. +– When comparing empirically obtained results in the T2/3- ˙Mt- +plane to our derived curves, we find a significant fraction of +stars to have lower values of ˙Mt than predicted by our curves +using hydrodynamic models. It is currently unclear whether +this is due to a deviation from the theoretical setup in this +work (e.g., different clumping stratification, other L-M com- +binations) or inherent simplifications in the empirical analy- +ses (e.g., the use of a β-type velocity law). A dedicated anal- +ysis of individual objects with hydrodynamical model atmo- +spheres will be necessary to uncover the origin of this dis- +crepancy. +– The limits of driving optically thick winds crucially de- +pend on our knowledge of opacities. In case of consider- +able changes – e.g., a higher iron opacity as reported by +Bailey et al. (2015) for the so-called “deep iron bump” at +Te ≈ 2 · 106 K – wind quantity predictions such as ˙M and �∞ +could shift significantly. Moreover, our understanding of the +limits of radiative driving would be affected as well, e.g., due +to strengthening or weakening the bumpy radius dependency +of the flux-weighted mean opacity κF. Beside the impact of +multi-D effects, higher (Fe) opacities could play an impor- +tant role to resolve current discrepancies, such as the lower +luminosity end of the LMC WN population or the aforemen- +tioned need for higher D∞ to reach the observed terminal +velocities. +With these conclusions, our study underlines the complexity +of radiation-driven mass loss, revealing both parameter regimes +with a clear scalings and characteristic transitions as well as +more obscure parameter regions where ˙M appears to break down +suddenly. We provide an adjustment of the recent ˙M-description +from Sander & Vink (2020) to account for different radii (or ef- +fective temperatures respectively) and emphasize that the model +efforts presented there as well as in this work were limited to the +regime where winds are launched by the hot iron bump. We thus +consider our work as an intermediate step on the way toward a +more comprehensive understanding of WR-type mass loss and +will expand to other regimes in future studies. +Acknowledgements. The authors would like to thank the anonymous referee for +their careful and constructive comments and suggestions. AACS and VR ac- +knowledge support by the Deutsche Forschungsgemeinschaft (DFG, German +Research Foundation) in the form of an Emmy Noether Research Group – +Project-ID 445674056 (SA4064/1-1, PI Sander). RRL is funded by the Deutsche +Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID +138713538 – SFB 881 (“The Milky Way System”, subproject P04). LP acknowl- +edges support by the Deutsche Forschungsgemeinschaft – Project-ID 496854903 +(SA 4046/2-1, PI Sander). JSV is supported by STFC funding under grant num- +ber ST/V000233/1. This publication has benefited from discussions in a team +meeting (PI: Oskinova) sponsored by the International Space Science Institute +(ISSI) at Bern, Switzerland. A significant number of figures in this work were +created with WRplot, developed by W.-R. Hamann. +References +Aadland, E., Massey, P., John Hillier, D., et al. 2022, ApJ, 931, 157 +Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2016, ApJ, 818, L22 +Bailey, J. E., Nagayama, T., Loisel, G. P., et al. 2015, Nature, 517, 56 +Bestenlehner, J. M. 2020, MNRAS, 493, 3938 +Bestenlehner, J. M., Crowther, P. A., Caballero-Nieves, S. M., et al. 2020, MN- +RAS, 499, 1918 +Björklund, R., Sundqvist, J. O., Puls, J., & Najarro, F. 2021, A&A, 648, A36 +Castor, J. I., Abbott, D. C., & Klein, R. 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Sander et al.: The temperature dependency of Wolf-Rayet-type mass loss +AS +-6 +-5 +-4 +5.0 +5.2 +5.4 +log(Teff(τcrit) [K]) +log( ˙M [M⊙ yr−1]) +Fig. A.1. Linear fits (solid lines) to the Mass-loss rate ˙M as a function +of Teff(τcrit) in a double-logarithmic-plane. The fit coefficients for the +different datasets are given in Table A.1. +Appendix A: ˙M-temperature Fit +For all of our model sequences, the data points in the log ˙M- +log Teff(τcrit)-plane suggest a linear relation between the two +quantities with deviations occurring only close to the wind driv- +ing limit (corresponding to the maximum ˙M in Fig. A.1). In the +fits, we thus exclude the uppermost 0.15 dex in log ˙M. The result- +ing fit coefficients for the slope including their error margins are +given in Table A.1. For each individual sequence, the mass loss +can be well described for Teff(τcrit) > Teff,crit,min and ˙M < ˙Mmax +by +log +� +˙M +M⊙ yr−1 +� += −6 log +� Teff,crit +141 kK +� ++ log +� +˙Moffset +M⊙ yr−1 +� +(A.1) +with the value for +˙Moffset being different for each model se- +quence. Table A.1 also lists these coefficients together with the +corresponding validity limits Teff,crit,min and ˙Mmax. +Appendix B: T2/3 temperatures for the Sander & +Vink (2020) sample +The effective temperatures T2/3 at τRoss = 2/3 resulting from +the model sequences in Sander & Vink (2020) are depicted in +Fig. B.1. Similar to what we obtain when varying T∗, cooler +values of T2/3 require higher mass-loss rates. As T∗ is fixed to +≈ 141 kK in Sander & Vink (2020), higher luminosities or L/M- +ratios are required to reach higher mass-loss rates for the same +Z. At lower metallicity, stars have to get closer to the Edding- +ton Limit to reach sufficient mass loss, shifting the onset of the +drop in T2/3 to higher L and steepening in particular this drop. +The differences in T2/3 and the range of luminosities covered has +quite some interesting implications on the spectral appearance of +the stars and consequently also which WR subtypes one would +expect in a certain environment. Assuming that at least all the +winds of early-type WR stars are launched at the hot iron bump, +the temperatures of the lowest luminosity WN stars should get +hotter at low Z. This seems to be the case when comparing the +WN populations in the Milky Way and the LMC (e.g., Hamann +et al. 2019; Shenar et al. 2020), but further – ideally hydrogen- +free – WN populations in other Galaxies need to be studied to +draw any firm conclusions. In the SMC, apart from the small +sample size, all WN stars contain hydrogen and might not align +4.4 +4.6 +4.8 +5.0 +log (T2/3 [kK]) +5.0 +5.5 +6.0 +6.5 +7.0 +log (L [L⊙]) +2.0 Z⊙ +1.5 Z⊙ +1.0 Z⊙ +0.5 Z⊙ +0.2 Z⊙ +0.1 Z⊙ +0.05 Z⊙ +0.02 Z⊙ +Fig. B.1. HRD with the effective temperature T2/3 defined at a Rosse- +land optical depth of τRoss = 2/3 and the model luminosity L for our +sets of He ZAMS models at different Z. For comparison, the HeZAMS +(gray, dashed) and ZAMS (gray, solid) are shown as well. +with the L-M relation we assume in Sander & Vink (2020) and +this work. +Appendix C: The effect of enhanced clumping on +the radiative acceleration +The effect of clumping on our hydrodynamic wind solutions is +not trivial. Given that we only use the so-called microclumping +approximation assuming optically thin clumps and the solution +of radiative transfer in the comoving frame is performed with +the average density and not the clumped density, one might ex- +pect that the choice of D∞ could have no effect at all. However, +this is not the case. Different values of D∞ affect the population +numbers which in turn affect the radiative transfer. In particular, +higher choices of D∞ favor recombination in the wind. For most +elements, lower ionization stages provide more opacity and thus +more radiative acceleration. +In Figs. C.1 and C.2 we present the resulting acceleration +contributions for the hydrogen-free 20 M⊙ WN models with +D∞ = 50 and 10, respectively. To obtain these curves, the in- +dividual opacities resulting from the different ions are stored in +addition to the total opacity. Beside the total calculation of +arad(r) = 4π +c +∞ +� +0 +κν(r) Hν(r) dν +(C.1) +similar integrals to Eq. (C.1) are calculated using only the ion- +specific opacities (e.g. κFe V +ν +) instead of the total κν, yielding the +specific acceleration contribution for each ion. The correspond- +ing wind parameters of the two displayed models and two other +sets with varying D∞ are listed in Table C.1. With out fixed char- +acteristic velocity for the clumping onset of �cl = 100 km s−1, +the depicted models increase from almost no clumping to D∞ +within the hot iron bump. Thus, the mass-loss rates are barely +affected, but the terminal velocity increases significantly from +1186 km s−1 to 1754 km s−1. +The comparison of Fig. C.2 with Fig. C.1 confirms that the +additional opacity to reach the higher terminal velocity is pro- +vided by lower ions, most notably Fe iv, which is the leading +accelerator in the outer wind for the model with D∞ = 50, while +Fe v remains in the lead for the model with D∞ = 10. In both +Article number, page 17 of 21 + +A&A proofs: manuscript no. paper +Table A.1. Linear fit results for log ˙M versus log Teff,crit plus offsets and limitations for the temperature-dependent mass loss of our model +sequences described by Eq. (A.1). +Sequence +slope +formal +log ( ˙Moffset [M⊙ yr−1]) +Teff,crit,min [kK] +log ( ˙Mmax [M⊙ yr−1]) +M [M⊙] +XH +Z [Z⊙] +D∞ +error +WN +20 +0.0 +1.0 +50 +−6.02 +0.03 +26.33 +92 +−3.60 +WN +20 +0.2 +0.5 +50 +−6.02 +0.05 +26.31 +88 +−3.51 +WN +12.9 +0.2 +1.0 +50 +−5.67 +0.06 +24.09 +94(br) +−4.20 +WN +15 +0.0 +1.0 +50 +−5.82 +0.07 +24.92 +105(br) +−4.36 +WC +20 +0.0 +0.5 +50 +−5.96 +0.05 +25.81 +118(br) +−4.60 +WN +20 +0.2 +1.0 +50 +−5.96 +0.04 +26.13 +98 +−3.62 +Notes. (br) For marked sequences, Teff,crit,min reflects the lower limit for winds driven by the hot iron bump. In all other sequences breakdown, this +values just refers to the minimum explored value. +Table C.1. Derived wind parameters for WN models with different D∞ +D∞ +log ( ˙M [M⊙ yr−1]) +�∞ [km s−1]) +log ( ˙Mt [M⊙ yr−1]) +WN, 20 M⊙, XH = 0, Z⊙ +10 +−4.42 +1186 +−3.77 +50 +−4.40 +1754 +−3.57 +WN, 20 M⊙, XH = 0.2, Z⊙ +4 +−4.34 +1194 +−3.89 +10 +−4.28 +1448 +−3.71 +50 +−4.28 +1970 +−3.50 +WN, 20 M⊙, XH = 0.2, 0.5 Z⊙ +4 +−4.44 +609 +−3.70 +10 +−4.35 +848 +−3.56 +50 +−4.28 +1394 +−3.35 +cases Fe v is the most populated Fe ion in the outermost wind, +while the “fresh” opacity provided by the lesser populated Fe iv +is most efficient for the line acceleration in the case of D∞ = 50. +In the case of D∞ = 10, the population of Fe iv instead is too low +to contribute significantly. The change in �∞ is further enlarged +by the significantly smaller deceleration zone in the D∞ = 50 +model. In the deeper wind layers, the higher clumping boosts the +contribution from the iron M-shell opacities and leads to an in- +creased bound-free contribution (i.e. recombination) from He ii. +The two other examples in Table C.1 illustrate that in some +cases also the mass-loss rate can be notably affected by changes +of D∞. In our models, this is a consequence of the fixed value +of �cl. For regimes where generally lower values of �∞ are +reached, e.g. in lower metallicity model set, often the whole +amount of acceleration is reduced, shifting also the region with +� ≈ 100 km s−1. This can then have two effects leading to a lower +˙M for lower values of D∞: first, a direct reduction of opacities in +the region that determines ˙M. In addition, the reduced wind den- +sity could push the star out of the regime where the critical point +is in a totally optically thick region (cf. Sander & Vink 2020), +which would lead to a further reduction in ˙M. +In the last column of Table C.1, we provide the resulting +transformed mass-loss rates ˙Mt. While there is already a scal- +ing with ˙M √D∞ in these, it does not compensate the clumping +changes as the square root of the D∞-ratios is much larger than +the changes in �∞ (and ˙M). For example, the hydrogen free mod- +els differ by √50/10 ≈ 2.24, while �∞ only increases by a factor +of 1.48. Therefore, the models with higher D∞ posses a higher +˙Mt, despite larger terminal velocities reducing its value. +Appendix D: Estimating effective temperatures in +stellar structure models +For stellar structure models, the occurrence of optically thick +winds usually spoils the straight-forward prediction of the ob- +servable effective temperature T2/3 (see, e.g., Groh et al. 2014, +for a more detailed discussion). In the previous Sect. 3.3, we ob- +tained that for a given clumping factor D∞, our model sequences +collapse almost perfectly to a single line in the ˙Mt-T2/3-plane, +yielding +T2/3 ∝ ˙M−1/2 +t +(D.1) +for log ( ˙Mt [M⊙ yr−1]) > −4.5, thereby providing us with a +potential path to predict the observable effective temperature. +The relation (D.1) seems to be approximately unaffected by +abundance (Xi) changes, but there is a clear offset for differ- +ent choices of D∞. In our calculations, there is a difference of +∆ log (T2/3 [K]) ≈ 0.08 . . . 0.1 between D∞ = 4 and D∞ = 10 and +∆ log (T2/3 [K]) ≈ 0.1 . . . 0.15 between D∞ = 10 and D∞ = 50, +but the current amount of data along the D∞ plane is insufficient +to provide a robust mathematical formula that could enable a +scaling with D∞. +A different slope of ≈ −2/3 was obtained in Sect. 3.3, when +considering the sample of Sander & Vink (2020) instead of our +new model sequences. The origin of the difference in the slopes +must be rooted in the different nature of the sequences: In the +new sequences calculated for this work, L and M are constant. +For higher mass-loss rates ˙M we then obtain lower values of �∞ +along a sequence. In the sequences from Sander & Vink (2020), +where we proceed to higher L/M-ratios along each dataset, such +a trend between ˙M and �∞ is only reached in the optically thin +part, while we obtained ˙M ∝ �∞ in the dense wind regime. As +a consequence, models from the two different sources with ap- +proximately the same value of ˙M will differ in their �∞. When +comparing the �∞-values, the models from the T∗-sequences in +this work will have lower terminal velocities and thus their ˙Mt +will be higher. Arguing that ˙M is the major factor in setting the +T2/3 value, we can thus conclude that this difference in the �∞ +trends leads to the steeper slope for the sequences along the +L/M-domain. +Given the focus of this work on the temperature trends +and the fact that the steep linear trends in Fig. 12 do not +provide a good description around the transition region of +log ( ˙Mt [M⊙ yr−1]) ≈ −4.5, we therefore suggest the more shal- +low formula +log (T2/3 [K]) = 2.9 − 0.5 log ( ˙Mt [M⊙ yr−1]) +(D.2) +as a first attempt to approximate the observable effective tem- +perature T2/3 of a WR star in stellar structure models, which is +Article number, page 18 of 21 + +A. A. C. Sander et al.: The temperature dependency of Wolf-Rayet-type mass loss +He I +He II +He III +C III +C IV +N II +N III +N IV +N V +O III +O IV +O V +Ne II +Ne III +Ne IV +Ne V +Ne VI +Ne VII +Ne VIII +Na III +Na IV +Na V +Na VI +Mg V +Mg VI +Al VI +Si IV +P IV +P V +S IV +S V +S VI +Cl IV +Cl V +CL VI +Ar III +Ar IV +Ar V +Ar VI +Ar VII +Ar VIII +K IV +K V +K VI +K VII +Ca III +Ca IV +Ca V +Ca VI +Ca VII +Fe IV +Fe V +Fe VI +Fe VII +Fe VIII +Fe IX +Fe X +Fe XI +Fe XII +Fe XIII +Fe XIV +Fe XV +Fe XVI +arad +apress +aThom +AS +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +-3 +-2 +-1 +0 +1 +2 +3 +4 +log (r/R∗ − 1) +log (a/g) +Fig. C.1. Contributions of the different ions to the radiative acceleration of a hydrodynamically consistent, hydrogen-free WN model with T∗ = +130 kK, log L/L⊙ = 5.7, M = 20 M⊙, and D∞ = 50: Different ions are denoted by a combination of different color and symbol. The total radiative +acceleration (arad), the Thomson acceleration from free electrons (aThom = Γe · g), and the contribution from gas (and turbulence) pressure (apress) +are also shown for comparison. The loosely dashed horizontal line denotes the total Eddington limit that needs to be overcome to launch a wind. +essentially a rounded version of Eq. (3). This formula implicitly +assumes D∞ = 50 and is recommended for ˙Mt > 10−4.5 M⊙ yr−1. +For lower estimates of D∞, the offset value of 2.9 would need to +be reduced by about 0.1 . . . 0.2. We emphasize that Eq. (D.2) is a +first approach that needs to be tested and likely refined in future +studies. +With Eq. (D.2) given, only the transformed mass-loss rate ˙Mt +needs to be known to determine T2/3. In Sander & Vink (2020), +we could show that for T∗ = 140 kK the quantity ˙Mt is practi- +cally independent of metallicity in the limit of optically thick +winds (“pure WR regime”). There, ˙Mt can be expressed as a +linear function of L/M with a possible deviation only occur- +ring for He stars with current masses above 50 M⊙. To check +whether this conclusion is independent of T∗, we calculate a +small set of models sequences with different L/M values for dif- +ferent Teff,crit ≈ T∗. The resulting trends are shown in Fig. D.1. It +is clear from Fig. D.1 that there is some uncertainty in the slopes +as well as a potential dependence of the slopes on T∗ itself, but +in general an approximately linear behavior is obtained for each +choice of Teff,crit ≈ T∗. Hence, we obtain a viable prediction +method for stellar evolution models. This method is particularly +elegant as it does not require any further assumptions about the +flux-weighted mean opacity or the shape of the velocity field as +for example necessary in the current wind-corrected tempera- +tures in the GENEC models (see, e.g., Groh et al. 2014). +To get a formula for +˙Mt that only depends on quantities +which can be obtained from stellar structure calculations, we can +use the result derived in Sect. 4.4. Considering that in the new +model sequences calculated for this work both L and D∞ are +constant within one sequence, we can conclude that ˙Mt ∝ ˙M/�∞ +and obtain +log ( ˙Mt [M⊙ yr−1]) = −7.2 · log (Teff(τcrit) [K]) + offset +(D.3) +or ˙Mt ∝ R3.6 +crit in the regime of optically thick winds, i.e. for +log ( ˙Mt [M⊙ yr−1]) > −4.5). +In a second step, we then merge Eq. (D.3), which has been +determined for sequences of constant L/M, with the L/M- +dependence obtained in Sander & Vink (2020). Together, we +synthesize the formula +log +˙Mt +M⊙ yr−1 = 1.25 log L/M +L⊙/M⊙ ++ 3.6 log Rcrit +R⊙ ++ ˙Mt,off(Xi, D∞). +(D.4) +Again, it might be more convenient to replace Rcrit with Teff,crit +and gauge this with the 20 M⊙ model at 141 kK. This then yields +Article number, page 19 of 21 + +A&A proofs: manuscript no. paper +He I +He II +He III +C III +C IV +N III +N IV +N V +O III +O IV +O V +Ne III +Ne IV +Ne V +Ne VI +Ne VII +Ne VIII +Na III +Na IV +Na V +Na VI +Mg V +Mg VI +Si IV +P V +S IV +S V +S VI +Cl IV +Cl V +Cl VI +Ar IV +Ar V +Ar VI +Ar VII +K IV +K V +K VI +Ca III +Ca IV +Ca V +Ca VI +Ca VII +Fe IV +Fe V +Fe VI +Fe VII +Fe VIII +Fe IX +Fe X +Fe XI +Fe XII +Fe XIII +Fe XIV +Fe XV +Fe XVI +arad +apress +aThom +AS +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +-3 +-2 +-1 +0 +1 +2 +3 +4 +log (r/R∗ − 1) +log (a/g) +Fig. C.2. Contributions of different ions to the radiative acceleration, plotted similar to Fig. C.1, but now for a model employing D∞ = 10. The +wind reaches a lower terminal velocity and the supersonic region with Γrad = arad/g < 1 is more pronounced than in the model with D∞ = 50. +AS +-5 +-4 +-3 +4.2 +4.3 +4.4 +4.5 +4.6 +log(L/M [L⊙/M⊙]) +log( ˙Mt [M⊙ yr−1]) +M∗ [M⊙] +13 +16 +20 +30 +50 +T∗ = 100 kK +T∗ = 110 kK +T∗ = 120 kK +T∗ = 130 kK +T∗ = 141 kK +T∗ = 150 kK +T∗ = 160 kK +Fig. D.1. Transformed mass-loss rate ˙Mt as a function of L/M for dif- +ferent sequences varying in T∗. All models use XH = 0.2 except the +dashed-dotted sequences having XH = 0 for comparison. Apart from the +red, dashed sequence (using D∞ = 10), all sequences employ D∞ = 50. +The gray, dotted line is an interpolation of the solid sequences along the +theoretical temperatures for the He ZAMS from Grassitelli et al. (2018) +and Langer (1989). The light dotted linear curves in the background +indicate the slope of 1.25 which is used in Eqs. (D.4) and onward. +log +˙Mt +M⊙ yr−1 = 1.25 log L/M +L⊙/M⊙ +− 7.2 log Teff,crit +141 kK − 9.39. (D.5) +In Eq. (D.5), we also dropped the offet ˙Mt,off(Xi, D∞) which con- +tains further, uncertain dependencies. These can alter the re- +sulting values of ˙Mt, e.g. by ≈ −0.2 dex when changing from +D∞ = 50 to 10. Inserting Eq. (D.5) into Eq. (D.2), we obtain the +final formula for estimating T2/3: +log +�T2/3 +K +� += 7.595 − 0.625 log L/M +L⊙/M⊙ ++ 3.6 log Teff,crit +141 kK, (D.6) +This formula is only valid for hydrogen-free WN stars as we +have considerable offsets for other chemical compositions in +˙Mt(L/M) (cf. Fig. D.1). While the conversion between ˙Mt and +T2/3 is unaffected by chemical composition (cf. Sect. 3.3), the re- +sulting radiative acceleration is not. For example, the additional +acceleration from free electrons in partially stripped stars with +remaining surface hydrogen leads to higher mass-loss rates than +in H-free stars of the same L/M-ratio (cf. Fig. 5), thereby sub- +stantially shifting the balance between ˙M and �∞ and the result- +ing ˙Mt-values. In a future study, we thus plan to extend Eq. (D.6) +by a hydrogen-dependent term. While various uncertainties, e.g., +of the precise slopes in ˙Mt(L/M) and T2/3( ˙Mt) limit the accu- +racy of Eq. (D.6), it is sufficient enough to tell whether observed +Article number, page 20 of 21 + +A. A. C. Sander et al.: The temperature dependency of Wolf-Rayet-type mass loss +Empirical results +MW H-free WN-s +MW H-free WN-w +LMC H-free WNs +SMC WRs +Model sequences +20 M⊙, XH = 0.0, Z⊙ +20 M⊙, XH = 0.2, Z⊙ +20 M⊙, XH = 0.2, 0.5 Z⊙ +12.9 M⊙, XH = 0.2, Z⊙ +15 M⊙, XH = 0.0, Z⊙ +20 M⊙, WC, 0.5 Z⊙ +Models with D∞ = 10 +20 M⊙, XH = 0.2, Z⊙ +12.9 M⊙, XH = 0.2, Z⊙ +AS +-5 +-4 +20 +60 +100 +140 +180 +220 +T∗ [kK] +log( ˙M [M⊙ yr−1]) +Fig. E.1. Analogous plot to Fig. 5, but now showing the mass-loss rate +˙M as a function of T∗ for our model sequences. +Model sequences +20 M⊙, XH = 0.0, Z⊙ +20 M⊙, XH = 0.2, Z⊙ +20 M⊙, XH = 0.2, 0.5 Z⊙ +12.9 M⊙, XH = 0.2, Z⊙ +15 M⊙, XH = 0.0, Z⊙ +20 M⊙, WC, 0.5 Z⊙ +AS +-6 +-5 +-4 +-3 +20 +60 +100 +140 +180 +220 +T∗ [kK] +log( ˙Mt [M⊙ yr−1]) +Fig. E.2. Analogous plot to Fig. E.1, but now showing the transformed +mass-loss rate ˙Mt instead of the normal ˙M. +effective temperatures of predicted objects are in the range of +e.g. 100, 50, or only 20 kK. Depending on the scientific con- +text, such differences in T2/3 can have a big impact. While we +do not have enough data to draw larger conclusions, the thick +gray dotted line Fig. D.1 illustrates that on the He ZAMS, com- +pact radii of the stars at the critical point might even outweigh +an expected increase in ˙Mt due to a larger L/M. Nonetheless, we +can present the first estimate of T2/3 for WN stars derived from +fundamental principles which can be readily applied in stellar +evolution models and population synthesis. The validity of the +assumptions made here will have to be tested within dedicated +test calculations in stellar evolution models and benchmarked +with WR observations analyzed with traditional as well as dy- +namically consistent models. +Appendix E: Additional Figures +In addition to Fig. 5 discussed in Sect. 3, we plot the mass-loss +rate ˙M as a function of T∗ in Fig. E.1. For very high mass-loss +rates, we see a bending of the curves toward lower values of T∗. +This is a consequence of the deeper wind launching, which in +this regime happens further in than the defining optical depth +for T∗ (i.e. at τR,cont > 20). Numerically, these models use a +higher optical depth as their inner boundary and we then deter- +Teff(τR = 2/3) +T∗(τR,c = 20) +Teff(τcrit) +Te(Rcrit) +AS +-6 +-5 +-4 +20 +60 +100 +140 +180 +220 +T [kK] +log( ˙M [M⊙ yr−1]) +Fig. E.3. Mass-loss rates as a function of different temperature scales for +a series of dynamically consistent atmosphere models with log L/L⊙ = +5.35, M = 12.9 M⊙, and XH = 0.2: The thick red dashed line denoted the +classical effective temperatures defined at a Rosseland optical depth of +τR = 2/3, while the green solid line and the blue dashed-dotted lines +denotes the effective temperatures referring to τcrit and τR,cont = 20 +respectively. The green dotted line on the right denotes the (electron) +temperature at the critical point. Curves in lighter colors reflect mod- +els using the simple integration treatment suppressing negative velocity +gradients. +Teff(τR = 2/3) +T∗(τR,c = 20) +Teff(τcrit) +Te(Rcrit) +AS +-6 +-5 +20 +60 +100 +140 +180 +220 +T [kK] +log( ˙M [M⊙ yr−1]) +Fig. E.4. Analogous plot to Fig. E.3, but now for the WC model series +with log L/L⊙ = 5.7, M = 20 M⊙, and 0.5 Z⊙. +mine T∗(τR,cont = 20) for a better comparison with the rest of +the model calculations. In this regime, T∗ is no longer a good +approximation for Teff,crit. +To eliminate the effect of different clumping factors and any +remaining distance uncertainties, it is helpful to consider the +transformed mass-loss rate +˙Mt instead of +˙M. In Fig. E.2, we +show the analogous plot to Fig. E.1 with ˙M being replaced by +˙Mt. While the vertical spread in the observations is slightly re- +duced, the general temperature mismatch between the empirical +T∗ and our model sequences remains, highlighting once more +the “Wolf-Rayet radius problem” seen in traditional atmosphere +analyses. +In an extend to Fig. 6 and Fig. 7, we show similar plots for +the model sequences with 12.9 M⊙, XH = 0.2 and Z = Z⊙ in +Fig. E.3 and the WC model sequence with 20 M⊙ and Z = 0.5 Z⊙ +in Fig. E.4. Similar to the result obtained for the 15 M⊙-sequence +discussed in Sect. 3.2, there is a lower minimum temperature for +obtaining wind solutions driven by the hot iron bump. +Article number, page 21 of 21 + diff --git a/SdAzT4oBgHgl3EQf0f6b/content/tmp_files/load_file.txt b/SdAzT4oBgHgl3EQf0f6b/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1479fbe2f542b7f2e3e6dba4c73cbdb2ca1bb441 --- /dev/null +++ b/SdAzT4oBgHgl3EQf0f6b/content/tmp_files/load_file.txt @@ -0,0 +1,2122 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf,len=2121 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' paper ©ESO 2023 January 6, 2023 The temperature dependency of Wolf-Rayet-type mass loss An exploratory study for winds launched by the hot iron bump A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Sander1, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Lefever1, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Poniatowski1, 2, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Ramachandran1, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Sabhahit3, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Vink3 1 Zentrum für Astronomie der Universität Heidelberg, Astronomisches Rechen-Institut, Mönchhofstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 12-14, 69120 Heidelberg e-mail: andreas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='sander@uni-heidelberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='de 2 Institute for Astronomy (IvS), KU Leuven, Celestijnenlaan 200D, 3000 Leuven, Belgium 3 Armagh Observatory and Planetarium, College Hill, BT61 9DG Armagh, Northern Ireland Received 3 October 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' accepted 27 December 2022 ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The mass loss of helium-burning stars, which are partially or completely stripped of their outer hydrogen envelope, is a catalyst of the cosmic matter cycle and decisive ingredient of massive star evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Yet, its theoretical fundament is only starting to emerge with major dependencies still to be uncovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' A temperature or radius dependence is usually not included in descriptions for the mass loss of classical Wolf-Rayet (cWR) stars, despite being crucial for other hot star wind domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' We thus aim to determine whether such a dependency will also be necessary for a comprehensive description of mass loss in the cWR regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Sequences of dynamically consistent stellar atmosphere models were calculated with the hydrodynamic branch of the PoWR code along the temperature domain, using different choices for the luminosity, mass, and surface abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For the first time, we allowed nonmonotonic velocity fields when solving the hydrodynamic equation of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The resulting velocity structures were then interpolated for the comoving-frame radiative transfer, ensuring that the main wind characteristics were preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' We find a strong dependence of the mass-loss rate with the temperature of the critical/sonic point which mainly reflects the different radii and resulting gravitational accelerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Moreover, we obtain a relation between the observed effective temperature and the transformed mass-loss rate ˙Mt which seems to be largely independent of the underlying stellar parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The relation is shifted when different density contrasts are assumed for the wind clumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Below a characteristic value of log ( ˙Mt [M⊙ yr−1]) ≈ −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5, the slope of this relation changes and the winds become transparent for He ii ionizing photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The mass loss of cWR stars is a high-dimensional problem but also shows inherent scalings which can be used to obtain an approximation of the observed effective temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For a more realistic treatment of cWR stars and their mass loss in stellar evolution, we recommend the inclusion of a temperature dependency and ideally the calculation of hydrodynamic structure models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' stars: atmospheres – stars: early-type – stars: evolution – stars: mass-loss – stars: winds, outflows – stars: Wolf-Rayet 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Introduction The mass loss of hot, evolved, massive stars plays a critical role on multiple astrophysical scales: strong stellar winds af- fect the individual appearance of the stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Hamann 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' de Koter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Hillier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Shenar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2020) and consequently also their ionizing and energetic feedback to the environment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Crowther & Hadfield 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Hainich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Sander & Vink 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Although the timescales for all evolutionary stages beyond the main sequence are comparably short, mass loss in these stages still consider- ably affects the stellar fates (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Langer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Chieffi & Limongi 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Yusof et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Moriya & Yoon 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In par- ticular for hydrogen-depleted, classical Wolf-Rayet (WR) stars, strong stellar winds provide a major channel for the chemical enrichment of their host environment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Maeder 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Dray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Farmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Martinet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The strong winds give rise to the emission-line-dominated spectra of WR stars, leaving their imprint even in integrated spectra of whole stellar populations and galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Conti 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Leitherer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Schaerer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Plat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Since the detectabil- ity of gravitational waves (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2016) and along with it the significant amount of black holes (BHs) above 20 M⊙ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', The LIGO Scientific Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2021), the interest in a better understanding of the BH-mass limiting WR mass loss as a function of metallicity (Z) has increased even further (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Woosley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Higgins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Vink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Contrary to their impact, the theoretical understanding of WR-type winds is still rather limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Despite earlier doubts, partially exacerbated by the high mass-loss rates determined be- fore clumping was incorporated into wind models, the consider- ations and model efforts of Nugis & Lamers (2002), Gräfener & Hamann (2005) and Vink & de Koter (2005) demonstrated that the winds of WR stars are mainly radiatively driven with iron opacities playing a critical role for the acceleration of the wind and the scaling of the mass-loss rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Since then, it required a new generation of computers and a considerable update to the modeling techniques to extend these fundamental efforts to a larger parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Only recently did Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2020) and Sander & Vink (2020) manage to calculate a larger set of dynamically consistent 1D atmosphere models that were able to predict the winds of classical WR (cWR) stars over a wider pa- rameter space, though still covering two dimensions only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' A key ingredient of these models is the detailed calculation of the flux- Article number, page 1 of 21 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='01785v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='SR] 4 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' paper weighted mean opacity κF and thus the radiative acceleration arad in an expanding environment without requiring the assump- tion of local thermodynamic equilibrium (LTE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The new gen- eration of computational capabilities has also opened the path toward multidimensional simulations for WR winds (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Poni- atowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Moens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In contrast to the 1D models, these 3D calculations are time-dependent, but so far lim- ited to LTE and very few test cases, making the current insights from 1D and 3D modeling quite complementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In this work, we mainly follow up on the work of Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2020) and Sander & Vink (2020), using 1D stellar atmo- sphere models to explore an additional dimension that is very important to determine the properties and strength of WR winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Given the high computational costs of dynamically consistent atmosphere models, all sequences presented in Sander & Vink (2020) were calculated using a fixed stellar temperature (T∗) de- fined at a Rosseland continuum optical depth of τR,cont = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The corresponding radii R∗ for the models were thereby given via the Stefan-Boltzmann law L = 4πR2 ∗σSBT 4 ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (1) While the value of T∗ = 141 kK in Sander & Vink (2020) was well motivated by the prototypical solution for a classical WC star (Gräfener & Hamann 2005), there is a priori no reason to assume that this choice of T∗ is valid for all He-burning WR stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In fact, stellar structure models predict a curvature in the zero age main sequence (ZAMS) for He stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Langer 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Köhler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2015) with lower temperatures obtained for lower masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Since we could not take this effect into account in Sander & Vink (2020), we had to limit the applicability of the derived ˙M recipe to He stars of about 10 M⊙ and higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The radii of WR stars and – as a consequence – also their temperatures are a long-standing topic of active research (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Hillier 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Hamann & Gräfener 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Grassitelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Beside the curvature in the He ZAMS, we are facing a particular challenge for stars with dense winds by the photosphere shifting to highly supersonic velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Thereby, the spectral appearance is completely determined in the wind, providing no direct observable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' log g) which could be used to determine the (hydrostatic) stellar radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This raises the prob- lem of connecting the effective temperatures for a Rosseland op- tical depth of τR = 2/3 (T2/3), which can be obtained via quanti- tative spectroscopy, to the (much) deeper subsonic regime repre- sented by T∗ for stars with extended envelopes (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' the discussion in Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In principle, the actual hydrostatic radii of the stars could be much larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This is referred to as (hydrostatic) inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Alternatively, a relatively compact star can be cloaked in a wind that is optically thick out to significant radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Both so- lutions might actually occur in nature with the realized branch depending on the particular stellar parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Stellar structure calculations can help to get a handle on R∗ and T∗ for helium stars, but their inherent (and computationally necessary) limitation to gray opacities usually prevents a proper estimation of T2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For a selected evolutionary track of a 60 M⊙ star, Groh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2014) calculated stellar atmosphere models adopting stellar parameters derived from an evolutionary track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This method led to important revisions on the predicted spec- tral appearances and their duration during the later evolutionary stages of massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' However, despite the more sophisticated method to obtain the improved effective temperatures, their un- derlying mass-loss rates were taken from a simplified recipe in- herent to the evolutionary calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' With the calculation of hydrodynamically-consistent atmo- sphere models, we can now obtain consistent mass-loss rates ˙M Rcrit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='023 R* R2/3 = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 R* raw dynamic solution solution with suppressed negative gradients interpolated solution for CMF-RT sound speed AS 0 200 400 600 800 3 2 1 0 1 2 3 4 5 log (r/R∗ − 1) �(r)[km s−1] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Illustrating example of the possible velocity treatments in case that the integration of the hydrodynamic equation of motions yields a nonmonotonic �(r) (solid blue curve): In the simple treatment, nega- tive velocity gradients are suppressed during the integration, yielding the blue, dash-dotted curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In some cases, such as illustrated in this plot, this can spoil the terminal velocity �∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In the more sophisticated treatment, negative gradients are therefore taken into account and �(r) is modified such that the interpolated solution keeps the obtained �∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' and effective temperatures for He-burning stars without requir- ing any prescription of ˙M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In this work, we use this technique to investigate the behavior of T2/3 and other temperature scales for multiple sequences of models with extended atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The paper is structured as follows: In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2, we briefly introduce the model atmosphere code including its recent updates neces- sary for our study as well as the calculated model sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 3, we present the resulting temperature trends, starting with an exemplary discussion of one sequence before exploring the full sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Afterwards, we take a closer look at the obtained trends in the terminal wind velocities in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Evolutionary implications and resulting scaling relations for the imprint of the WR effective temperatures are discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The insights on He ii ionizing fluxes are introduced in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 6 before drawing the conclusions in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Stellar atmosphere models In this work we employ the PoWR model atmosphere code (Gräfener et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Hamann & Gräfener 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2015) in its hydrodynamical branch (PoWRHD) to calculate sta- tionary, hydrodynamically-consistent atmosphere models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The implementation concepts for coupling hydrodynamics and ra- diative transfer are described in Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2017, 2018) and Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Our hydrodynamic solutions are calculated in a similar manner as described in Sander & Vink (2020), that is we keep the stellar parameters L and M∗ fixed and iteratively adjust ˙M and �(r) until a consistent solution is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In our previous studies, the solutions obtained for the veloc- ity field were always monotonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In this work, we demonstrate that apart from the high clumping factor (D∞ = 50), this was mainly a result from choosing T∗ = 141 kK as the anchor point of our model sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' When extending our modeling efforts to lower T∗, we now approach a regime where Γrad := arad/g can drop below unity after launching the wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Such a deficiency in the available ra- diative acceleration is regularly seen in WR atmosphere mod- els including the opacities of the hot iron bump (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Gräfener Article number, page 2 of 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' : The temperature dependency of Wolf-Rayet-type mass loss Rcrit R2/3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1 2 10 100 1000 r/R∗ no negative dv/dr in HD sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' g + amech arad + apress interpolation after HD sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' g + amech arad + apress AS 1 0 1 2 1 0 1 2 3 log (r/R∗ − 1) log (a/g) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1 2 10 100 1000 r/R∗ 400 800 1200 2 1 0 1 2 3 log (r/R∗ − 1) v(r) [km s−1] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Resulting acceleration stratification of the converged models with two different treatments of nonmonotonic velocity fields: The dashed red and black lines illustrate the two sides of the hydrodynamic equation of motion for a model where negative velocity gradients are ignored in the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The corresponding solid lines show the re- sult for the alternative method where the nonmonotonic �(r) is inter- polated afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The small inlet shows the resulting velocity fields for both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In this example we show models with T∗ = 125 kK, log L/L⊙ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='475 and M = 15 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' & Hamann 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Aadland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' When assuming a pre- scribed velocity field, as traditionally done in spectral analysis, these deficiency regions have no immediate impact on the model calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This is different in our case where we solve hydro- dynamic equation of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Here, a region with Γrad < 1 in the supersonic regime implies a negative velocity gradient until Γrad eventually raises above unity again, resulting in a nonmonotonic velocity �(r) (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', the recent calculations and discussions in Poniatowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Given that our atmosphere modeling technique needs to perform the radiative transfer in a co-moving frame, which cannot handle nonmonotonic velocity fields, we therefore have to modify the �(r) obtained from the solution of the hydrodynamic equation of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Two approaches are used in this work, which are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In the simple, numerically more robust method, we ig- nore any negative gradients already during the solution of the equation of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' With this method, we obtain a locally con- sistent solution at all depth points except for any supersonic re- gions where Γrad < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' However, this method leads to an over- prediction of �∞ as any reduction in �(r) due to parts with neg- ative gradients is ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The example with the dashed-dotted curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 1 illustrates that this approach can remove all fur- ther structure from the outer velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' We thus calculate a second type of models where the integration of the hydrody- namic equation of motion is not perturbed and only the result- ing velocity field is interpolated afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For the latter inter- polation, we use an “outside-in” approach, starting at the outer boundary of our model (Rmax) and cutting away any parts where �(r) increases inward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' While the such modified solution actually leads to more local violations of the hydrodynamic equation of motion (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2), we preserve not only the correct �∞ but usu- ally also the whole �(r) in the optically thin regime as illustrated with the red-dashed curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' We therefore use these types of models as the main anchor-point for discussing our results and drawing conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Input parameters for our hydrodynamically consistent He- ZAMS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The absolute mass fraction for a particular model se- quence can be obtained by obtained by inserting the corresponding value from Table 2 for Z/Z⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Parameter Value(s) T∗ [kK] 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='220 log (L [L⊙]) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='35, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='475, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='7 M∗ [M⊙] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9, 15, 20 �mic [km s−1] 30 D∞ 10 or 50 abundances in mass fractions: XH 0 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 XHe 1 − XH − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='014 · Z/Z⊙ XC 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='7 · 10−5 · Z/Z⊙ XN 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1 · 10−3 · Z/Z⊙ XO 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 · 10−5 · Z/Z⊙ XNe 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='3 · 10−3 · Z/Z⊙ XNa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='7 · 10−6 · Z/Z⊙ XMg 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9 · 10−4 · Z/Z⊙ XAl 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='3 · 10−5 · Z/Z⊙ XSi 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 · 10−4 · Z/Z⊙ XP 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='8 · 10−6 · Z/Z⊙ XS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1 · 10−4 · Z/Z⊙ XCl 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 · 10−6 · Z/Z⊙ XAr 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='3 · 10−5 · Z/Z⊙ XK 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1 · 10−6 · Z/Z⊙ XCa 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1 · 10−5 · Z/Z⊙ XFe 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='6 · 10−3 · Z/Z⊙ Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Overview of the calculated model sequences type M [M⊙] log (L [L⊙]) XH Z [Z⊙] D∞ Main sequences WN 20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 50 WN 20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 50 WN 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 50 WN 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='475 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 50 WC 20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 50 WN 20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 50 Comparison sequences WN 20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 10 WN 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 10 WN 20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 4 To be able to compare our results to the large set of calcula- tions performed in Sander & Vink (2020), we keep most of our original model input, including the clumping description (D∞ = 50 and �cl = 100 km s−1, using the “Hillier law” from Hillier & Miller 1999) and the set of considered elements (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' However, we have calculated some additional models, includ- ing two complete T∗ sequences for 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9 M⊙ and 20 M⊙, with D∞ = 10 as well as one sequence with D∞ = 4 to have a com- parison sets which turns out to be quite insightful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 3 we display the resulting contributions to the ra- diative acceleration from two models which only differ in D∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The higher D∞ changes the wind stratification, most notably by stronger recombination from He iii to He ii (indicated by the higher He ii bound-free opacity bump) and an earlier ioniza- tion change from Fe vi to Fe v in the outer wind, enabling ad- Article number, page 3 of 21 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' paper bound-free opacities: He I He II He III line opacities: C N O Ne Si P S Cl Ar K Ca Fe IV Fe V Fe VI Fe VII Fe VIII Fe IX Fe X Fe XI Fe XII Fe XIII Fe XIV Fe XV Fe XVI arad apress aThom AS D∞ = 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 log (a/g) AS D∞ = 50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 3 2 1 0 1 2 3 4 log (r/R∗ − 1) log (a/g) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Major contributions to the radiative acceleration for two hydrodynamically consistent, hydrogen-free WN models with T∗ = 130 kK, log L/L⊙ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='7, M = 20 M⊙, and D∞ = 10 (upper panel) or D∞ = 50 (lower panel): For the line contributions, all elemental contributions except Fe are summed over all ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The total radiative acceleration (arad), the Thomson acceleration from free electrons (aThom = Γe · g), and the contribution from gas (and turbulence) pressure (apress) are also shown for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The loosely dashed horizontal line denotes the total Eddington limit that needs to be overcome to launch a wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' ditional line driving from Fe iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (A more detailed breakdown with all ionic contributions is provided in appendix Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' with Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 and brief discussion about the comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=') Furthermore, we calculate a few additional sequences where we add surface hydrogen (XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2) and one sequence where we switch to a WC-type (XC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='4, XN = 4 · 10−5, XO = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='05) metal composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' A full overview of the model sequences is given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Similar to Sander & Vink (2020), we again align L and M∗ such that they follow the relations for hydrogen-free stars given in Gräfener et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This means that we ignore any potential extra mass (and luminosity) due to surface hydrogen as well as any differences in the L-M relation between WN and WC stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' We do so in order to isolate the effects of different chemical compositions on the resulting wind predictions rather than trying to emulate an observed star or a particular evolutionary model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' A more detailed investigation of the impact of (surface) hydrogen on the mass loss of WR stars is currently underway and will follow in a separate paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' While not tailored to mimic any particular WR star, the spec- tra resulting from our model sequences display a typical WR- type appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' As an example, we plot four spectra from the 20 M⊙ WN sequence with XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z = Z⊙ and D∞ = 10 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The hottest model shown mimics typical features of a rel- ative weak-lined, early-type WN with intrinsic absorption lines, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', seen in WN3ha stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Along the sequence, the lines tend to get stronger, but narrower with additional lines from cooler ion- ization stages appearing in the cooler models that would be clas- sified as later WN types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Fine-tuned model efforts for particular stars will be necessary to further constrain choices of currently free parameters such as microturbulence and clumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Ideally, one would want to eliminate the necessity for a dedicated in- put of these parameters completely to get full dynamical consis- Article number, page 4 of 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' : The temperature dependency of Wolf-Rayet-type mass loss Teff,crit = 127 kK, T2/3 = 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='3 kK, log gcrit = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='41 log ˙M = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='28 Teff,crit = 148 kK, T2/3 = 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 kK, log gcrit = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='67 log ˙M = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='67 Teff,crit = 168 kK, T2/3 = 135 kK, log gcrit = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='89 log ˙M = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='99 Teff,crit = 187 kK, T2/3 = 170 kK, log gcrit = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='08 log ˙M = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='29 AS 1 3 5 7 9 4000 5000 6000 7000 λ [Å] normalized flux + offset Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Example spectra from our WN model sequence with log L/L⊙ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='7 and M = 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z = Z⊙, and D∞ = 10 for the optical wavelength regime with different colors corresponding to the different models tency, but this would require significant code updates, which is beyond the scope of the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Temperatures and radii For our sequences listed in Table 2, we study the mass-loss rate as a function of the effective temperature at the critical (≈ sonic) point Teff(τcrit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The latter value is very close to T∗ defined at τR,cont = 20, which usually describes the inner boundary of our models, except for models with very high ˙M where τR,cont = 100 needs to be chosen as the inner boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The results depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 5 reveal a steep, monotonic decrease of ˙M with increasing value of Teff(τcrit) (corresponding to decreasing radii Rcrit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This is not unexpected given that larger radii lower the local gravita- tional acceleration and thus enable an easier escape of material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Our sequences with different chemical compositions give us first qualitative insights on the impact of hydrogen on the one hand and carbon and oxygen on the other hand: The direct com- parison of the two curves for 20 M⊙ at Z⊙ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 5 show a systematic shift to slightly higher mass-loss rates in the pres- ence of surface hydrogen (as long as it is negligible for the to- tal stellar mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Interestingly, a hydrogen surface mass fraction of XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 seems to be sufficient to counter the lower metal abundances in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This result should not yet be generalized given the limited number of sequences, but will be followed up in our dedicated study focusing on surface hy- drogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Contrary to hydrogen, the inclusion of more carbon and oxygen is not beneficial to ˙M as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 5 by the se- quence with the WC surface composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In all cases, the local Empirical results MW H-free WN-s MW H-free WN-w LMC H-free WNs SMC WRs Model sequences 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0, Z⊙ 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ 15 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0, Z⊙ 20 M⊙, WC, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ Models with D∞ = 10 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ AS 5 4 20 60 100 140 180 220 Teff(τcrit) [kK] log( ˙M [M⊙ yr−1]) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Mass-loss rate ˙M as a function of Teff(τcrit) for our model se- quences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For comparison, a set of empirically inferred temperatures T∗ for different types of WR stars is shown as well (gray symbols), illustrat- ing the well-known “WR radius problem.” Since the empirical values are inferred from models without dynamical consistency, the values of T∗, defined at a Rosseland optical depth of τR,cont = 20, are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For our model sequences, the values of T∗ and Teff(τcrit) align very closely except for the highest mass-loss rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' A direct comparison with T∗ from the dynamically-consistent models is provided in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (electron) temperatures at the launching point of the wind (Rcrit) are too high to generate any additional line opacity from C or O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Article number, page 5 of 21 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' paper Teff(τR = 2/3) T∗(τR,c = 20) Teff(τcrit) Te(Rcrit) Grassitelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2018) Teff(τs) AS 6 5 4 20 60 100 140 180 220 T [kK] log( ˙M [M⊙ yr−1]) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Mass-loss rates versus different temperature scales for a series of dynamically consistent atmosphere models with log L/L⊙ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='7, M = 20 M⊙, and XH = 0: The thick red dashed line denotes the ef- fective temperatures defined at a Rosseland optical depth of τR = 2/3, while the green solid line and the blue dashed-dotted line denote the ef- fective temperatures referring to τcrit and τR,cont = 20, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The green dotted line on the right denotes the (electron) temperature at the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Curves in lighter colors reflect models using the simple in- tegration treatment suppressing negative velocity gradients (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The black dashed line shows the hydrodynamic structure solutions by Grassitelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Instead, the slightly lower amount of free electrons compared to a WN surface composition decreases the resulting ˙M (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The opposite effect instead happens in the case of WN stars with XH > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For comparison, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 5 also contains empirical results ob- tained with standard models using a β-law or double-β-law to describe �(r) from Hamann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2006, 2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Hainich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Shenar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2016, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Illustrating the well-known “Wolf-Rayet radius problem” (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Grassitelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2018), most empirically derived temperatures are located at T∗ values that seem to be too cool for their mass-loss rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' As demonstrated by Gräfener & Hamann (2005) and Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2020), the critical radii inferred from dynamically-consistent atmosphere models are much smaller than those obtained by a standard β-law due to the opacities of the “hot iron bump” that enable the launch of a supersonic wind already at deeper layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Moreover, the com- parison in the ˙M-Teff(τcrit) plane is not ideal as the empirically derived values of ˙M depend on distances which are still uncer- tain for some Galactic targets, but as we see below when dis- cussing the transformed mass-loss rate, this is not a major issue here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' An inspection of the model sequences indicates a clear shift for different mass regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Thus, one could argue that the whole plane in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 5 could be covered if we would calculate further sequences for lower masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' However, lower masses are most likely not the solution and discrepancies remain even when ad- justing the mass-loss rates for stars with different luminosities in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Mass loss versus different temperature scales To discuss the different temperature scales in WR winds and their scaling with ˙M, we take a closer look at an individual model sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 6, we plot the mass-loss rate ˙M for the se- quence of 20 M⊙ models without hydrogen as functions of differ- ent temperature definitions, namely (i) the effective temperature T2/3 at a Rosseland optical depth of τR = 2/3 (thick red dashed line), often simply denoted as Teff in the literature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (ii) the ef- Teff(τR = 2/3) T∗(τR,c = 20) Teff(τcrit) Te(Rcrit) Grassitelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2018) Teff(τs) AS 6 5 4 20 60 100 140 180 220 T [kK] log( ˙M [M⊙ yr−1]) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 6, but for a model sequence with log L/L⊙ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='475, M = 15 M⊙, and XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Contrary to the situation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 6 for a 20 M⊙ star, there is an abrupt breakdown of solutions beyond a minimum tem- perature and a maximum ˙M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For the T2/3− and Teff(τcrit)-scales these points are marked with a vertical line attached to a gray-hatched area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' fective temperature T∗ commonly used in the model setup for PoWR models, defined at a Rosseland continuum optical depth of τR,cont = 20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (iii) the effective temperature at the critical point, denoted Teff(τcrit), Teff(Rcrit), or simply Teff,crit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' and (iv) the elec- tron temperature Te at the critical point (thin green dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In contrast to the first three temperatures, Te is not an effective temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In the deeper layers of the atmosphere where the deviations from LTE become negligible, Te aligns with the gen- eral temperature T(r) defined in (1D) stellar structure models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' To further illustrate the effect of the two different nonmonotonic �(r)-treatments, we plot the more accurate method with posterior interpolation of the velocity field in strong colors while the sim- ple method ignoring negative gradients is drawn in lighter shades of the same line style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Beyond a certain temperature, there are no more deceleration regions and thus both curves agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Except for the regimes with highest mass-loss ( ˙M > 10−4 M⊙ yr−1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 6), the values of T∗ and Teff,crit closely align.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This does not imply that τcrit has to correspond to τR,cont ≈ 20, but that the locations of the radii corresponding to τR,cont = 20 and τcrit are close enough to yield similar effective temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The alignment between T∗ and Teff,crit at lower ˙M is fulfilled in all of our model sequences (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 7 and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='3 for further exam- ples) and allows us to discuss the physically more meaningful temperatures at the critical point (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', the launch of the wind) in- stead of the slightly more technical T∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For higher ˙M, τcrit moves inward and eventually surpasses τR,cont = 20, explaining the de- viation of the curves for the highest mass-loss rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' However, these cases require models very close to the Eddington limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The growing difference between the effective temperature at the launch of the wind (Teff,crit) and T2/3 with increasing ˙M shows the “extended atmosphere” of a WR star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For low mass- loss rates, the atmosphere is optically thin and the two tempera- tures align.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Although at usually much lower temperatures, this is similar to what we see for most OB-star winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' With increasing ˙M, we get a more and more extended optically thick layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Al- beit leading to a much cooler appearance of the star, this kind of layer should not be mixed up with the inflated envelope obtained in various hydrostatic structure models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Petrovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Gräfener et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Ro & Matzner 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Instead of a subsonic, but still loosely bound extended layer, our models show super- sonic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', unbound) layers moving out with hundreds of km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' As illustrated in Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2020), the winds often reach more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 �∞ before the atmosphere becomes optically thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In more recent work (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Poniatowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2021), this form of Article number, page 6 of 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' : The temperature dependency of Wolf-Rayet-type mass loss 50 100 150 200 250 300 350 T [kK] −7 −6 −5 −4 −3 log ( ˙M [M⊙ yr−1]) Teff(τR = 2/3) Teff(τcrit) Te(Rcrit) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 Z⊙ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 Z⊙ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 Z⊙ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1 Z⊙ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='05 Z⊙ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='02 Z⊙ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Mass-loss rates as a function of different temperature scales for the L/M model sequences presented in Sander & Vink (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Higher mass-loss rates correspond to higher L/M-ratios in this plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' an extended photosphere is termed “dynamical inflation” to dis- tinguish it from the hydrostatic inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Hydrostatic inflation is not likely to occur if a wind can be launched and maintained, but it could in situations where the latter is not given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, we discuss the limits of launching a wind from the hot iron bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' While a detailed exploration of wind solutions beyond this limit is not feasible in this study, the numerical results of our “failed” models indicate a tendency toward larger sonic radii, potentially indicating some form of hydrostatic inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Finally, we also plot the electron temperature at the criti- cal (≈ sonic) point as a dotted green curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The val- ues reflect the expected temperature range of the hot iron bump (around 200 kK), although the particular values are slightly higher than predicted in structural studies employing OPAL opacity tables (Grassitelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Nakauchi & Saio 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The value of Te(Rcrit) appears relatively constant at first sight, but aside from numerical scatter affecting the results a bit, a sub- tle trend can be noticed: In the regime of optically thick winds, there is a tendency toward increasing Te(Rcrit) with higher mass- loss rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This is qualitatively in line with the predictions by Grassitelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2018), who found that in hydrodynamic stel- lar structure calculations higher mass-loss rates correspond to higher temperatures at the sonic point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' However, this trend is in- terrupted when the winds become optically more thin and even- tually Te(Rcrit) increases mildly with lower ˙M until Teff,crit sur- passes Te(Rcrit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' All of the temperature trends described for the exemplary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 6 are observed for the other model sequences as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For comparison, we show similar temperature scale plots for the 15 M⊙ sequence (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 7) and the 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9 M⊙ sequence with surface hydrogen (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' While there are shifts in the absolute values, the same general trends are clearly identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' To study whether our findings are more general or limited to our new sample, we check the behavior of the different tem- perature scales also for the whole set of model sequences from Sander & Vink (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The resulting curves are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Due to the fixed value of T∗(τR,cont = 20) = 141 kK in Sander & Vink (2020) and the launching of the winds at high optical depths, the effective temperature referring to the critical point (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', the launch of the wind) hardly varies over the whole sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Still, the resulting T2/3-temperatures look very similar to those obtained in our new models with varying T∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' On the other hand, Te(Rcrit) varies much more than in any of our new model R2/3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1 2 10 100 1000 r/R∗ arad acont apress athom AS 1 0 1 2 1 0 1 2 3 log (r/R∗ − 1) log (a/g) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Illustration of the radiative acceleration for a model with 10 M⊙ and T∗ ≈ 115 kK which is not capable of launching a wind from the “hot iron bump” and thus is dynamically not converged: In the inner part, the total radiative acceleration (red dashed line) approaches Γrad = 1, but does not surpass it sufficiently to launch a wind that could be maintained in the following deceleration region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' As we also see a shift in the Te(Rcrit)-curves between different mass sequences in our new work, we can conclude that Γe – defined by the chemical composition and L/M – plays a ma- jor role in setting the temperature regime of the sonic point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The ratio between the flux and the radius – which is mapped in T∗ – instead only has a minor effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' We do not see the interruption of the Te(Rcrit)-trend in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 8 that was apparent in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 6 and the other new model sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This is likely due to the different dimensionality of the sequences in Sander & Vink (2020) (fixed T∗, variable L/M per sequence) and this work (fixed L/M, vari- able T∗ per sequence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In general, we can conclude that for stars further away from the Eddington Limit, the same ˙M can only be reached by shifting the critical point to lower electron temper- atures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For the same L/M-ratio, however, we see a much lower amplitude of changes in Te(Rcrit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In a zeroth-order approxima- tion, one could state that Te(Rcrit) is constant for a given L/M and chemical composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' WR-type mass loss and its breakdown The trend of increasing ˙M with lower Teff,crit does not automat- ically continue beyond the plotted values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The comparison be- tween the simpler and the more sophisticated treatment in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 7 already suggests that the effect of deceleration regions has to be taken into account for computing a more realistic ˙M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In some situations, such as the one illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 7, the deceleration region can become large enough to reduce the wind to subsonic or even negative velocities, making it impossible to launch a wind from the deeper layers of the “hot iron bump.” This regime occurs right next to the (theoretical) maximum of ˙M along the Teff,crit-axis which is reached when the deceleration region is just not strong enough to put �(r) below the local sound speed in the wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The situation of a failed wind launch is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 9, where the radiative acceleration barely reaches Γrad = 1 in a model for 10 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This example also illustrates that for lower L/M values, the regime where no wind can be launched from the hot iron bump gets larger and larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' A hydrogen-rich sur- Article number, page 7 of 21 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' paper Model sequences 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0, Z⊙ 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ 15 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0, Z⊙ 20 M⊙, WC, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ Models with D∞ = 10 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ Models with D∞ = 4 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ AS 5 4 3 2 20 60 100 140 180 220 T2/3 [kK] log( ˙Mt [M⊙ yr−1]) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Transformed mass-loss rate as a function of T2/3 for our model sequences face can compensate this to some degree as it helps to get the star closer to the Eddington limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Still, when getting to lower and lower L/M-ratios the regime of WR winds driven by the hot iron bump eventually vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In our models with a fixed L-M- relation this corresponds to a limit in both luminosity and mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' However, objects of lower masses and luminosity can potentially launch a wind if they have a considerably higher L/M-ratios than homogeneous He stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This might e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' be the case in WR-type central stars of planetary nebulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Gräfener et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2017) and Ro (2019) also pointed out that most of the H-free WN population in the LMC presents a challenge as these stars should not be able to launch a wind from the hot iron bump if their masses would adhere to a typical L-M relation for He-burning stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' However, this discrepancy is already reduced if we use the Sander & Vink (2020) models, likely due to the computed flux-weighted opaci- ties exceeding the OPAL Rosseland opacities assumed as a proxy for κF in previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The discrepancy could potentially be reduced even further slightly lower temperatures are considered as well (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Nonetheless, the LMC sample remains an interesting test-bed for detailed comparisons with individual ob- jects and the limits of radiation-driven winds from the hot iron bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Summarizing the limits of ˙M along the temperature axis, we see two very different behaviors: Toward cooler temperatures, we have an abrupt breakdown of the thick wind regime when the effect of the deceleration region outweighs the initial accelera- tion by the hot iron bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This endpoint is reached close to the highest possible mass-loss rate (for the given stellar parameters) in this whole wind regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' On the hot temperature end, we in- stead proceed rather smoothly into the regime of optically thin winds with lower and lower values of ˙M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This drop along the T- axis is significantly shallower than the strong breakdown of ˙M along the L/M-axis we obtained in Sander & Vink (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Scaling with the transformed mass-loss rate Since the different calculated model sequences show very sim- ilar slopes for ˙M(T2/3), we investigate whether there is a com- mon scaling behind these curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Given the empirical scaling relations for WR spectra and our findings from Sander & Vink (2020), we study the “transformed mass-loss rate” ˙Mt = ˙M √ D · �1000 km/s �∞ � �106L⊙ L �3/4 , (2) 10 20 50 100 200 T2/3 [kK] Empirical results MW H-free WN-s (Hamann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2006, 2019) MW H-free WN-w (Hamann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2006, 2019) MW WCs (Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2012, 2019) LMC H-free WNs (Shenar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2019) LMC WCs & WOs (Aadland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2022) SMC WRs (Hainich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2015, Shenar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2016) Model sequences 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0, Z⊙ 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ 15 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0, Z⊙ 20 M⊙, WC, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ Models with D∞ = 10 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ Models with D∞ = 4 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ AS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='3 5 4 3 2 log( ˙Mt [M⊙ yr−1]) log(T2/3 [K]) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Effective temperature at a Rosseland optical depth of 2/3 as a function of the transformed mass-loss rate ˙Mt for our new calculated model sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The sequences connected by solid lines all employ D∞ = 50, while those with dashed and dotted curves indicate sequences using D∞ = 10 and 4, respectively, as indicated in the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For compar- ison, also various empirical results from the literature are depicted by discrete, gray symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' originally introduced by Gräfener & Vink (2013), for our new model sequences as a function of T2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 10, the resulting curves align extremely well when plotting ˙Mt in- stead of ˙M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Major offsets are only introduced when assuming different (maximum) clumping factors D∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Given our findings in Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2020) and Sander & Vink (2020), the latter is not much of a surprise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' While the clumping does not directly affect the radiative transfer, the solution of the statistical equa- tions are solved for a higher density D · ρ (Hamann & Koesterke 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This affects the ionization stratification and usually leads to a larger opacity and thus larger terminal velocity �∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' While the mass-loss rate ˙M is typically not much affected, the resulting ˙Mt is changed due the increase in �∞ being smaller than the in- crease in √D∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For our 20 M⊙ models with XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, the typical increase was about 20% in �∞ when increasing D∞ from 4 to 10 and about 40% when increasing D∞ from 10 to 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' To quantify our finding, we flip the axes and show a double- logarithmic plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' A clear transition between two regimes is evident with a “kink” around log ˙Mt ≈ −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 that ap- pears to be independent of D∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The more dense wind regime (T2/3 < 130 kK and log ˙Mt > −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5) can be reasonably well ap- proximated by a linear fit, yielding log T2/3 K = (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='01) log ˙Mt M⊙ yr−1 + (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='02) (3) for the sequences using D∞ = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Given the inherent numerical scatter, in particular in �∞ entering ˙Mt, we can conclude that in the limit of dense winds T2/3 ∝ ˙M−1/2 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' When comparing the relations with empirically obtained val- ues of WN (Hamann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2006, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Hainich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Shenar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2016, 2019) and WC stars (Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2012, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Aadland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2022), it is immediately evident that comparing T2/3 between empirical and theoretical results yields a much bet- ter match than the comparison between the empirical T∗ and our theoretical Teff(τcrit) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 5 (or the direct T∗-comparison in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' While the mismatch in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 5 illustrates the “Wolf- Rayet radius problem” discussed at the beginning of Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 3, the better alignment of the T2/3 values underlines that value of the empirical analysis, despite the dynamical concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In empirical studies with fixed velocity fields, models are chosen such that Article number, page 8 of 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' : The temperature dependency of Wolf-Rayet-type mass loss −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 log ( ˙Mt [M⊙ yr−1]) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 log (T2/3 [K]) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 Z⊙ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 Z⊙ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 Z⊙ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1 Z⊙ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='05 Z⊙ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='02 Z⊙ L/M seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' trend T seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' trend Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Effective temperature at a Rosseland optical depth of 2/3 as a function of the transformed mass-loss rate ˙Mt for the whole set of models from Sander & Vink (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For log ( ˙Mt [M⊙ yr−1]) < −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5, T2/3 is effectively independent of ˙Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In Sander & Vink (2020), the value of T∗ is fixed for all models, but the difference in L/M still yields a wide range of T2/3 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The dashed-dotted line represents a linear fit of the temperature trend for log ( ˙Mt [M⊙ yr−1]) > −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 while the dotted curve represents the fit for the new model sequence illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' they reproduce the observed spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Although τ2/3 has a sig- nificant wavelength dependence in WR winds, the effective tem- perature corresponding to the Rosseland mean value provides some form of a representative value for the regime that needs to be met when the light eventually escapes from the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Our dy- namically consistent models can generally reproduce these T2/3 values, but employing more compact radii that better align with structural predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Despite the generally better match when comparing T2/3, it is also evident from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 11 that all symbols are either on or left- ward of the derived curve for D∞ = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The most striking dis- crepancies are obtained for the SMC WN stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The Aadland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2022) WC and WO results are very close to our obtained relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' As they are the only one assuming D∞ = 20, some dis- crepancies are likely rooted in different clumping assumptions and treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The mismatch of the empirical SMC positions however, cannot be explained with clumping differences alone with most of the stars showing empirical ˙Mt values that are about an order of magnitude lower than our model relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This could be due to various effects including considerable differences in L/M, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', due to having significant hydrogen shells and thus not obeying the assumed L-M-relation in our model sequences, or too low T2/3 estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Investigating these and other possibilities would add further dimensions to our model sequences and thus we have to postpone a dedicated analysis of individual targets to a separate follow-up paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' When considering the obtained curves in the T2/3- ˙Mt-plane from the current model sequences, it is so far unclear whether the slope and even the underlying scaling is universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 12, we thus plot the same parameters, now using the sequences from Sander & Vink (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' A first noticeable difference is the upper horizontal cutoff at T2/3 ≈ 141 kK, but this is expected due to the fixed value of T∗ = 141 kK in the Sander & Vink (2020) sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The behavior at higher values of ˙Mt looks similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 11 at first – although with considerably more scatter – but an actual fit of the data reveals a non-negligible difference in the slopes, Model sequences 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0, Z⊙ 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ 15 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0, Z⊙ 20 M⊙, WC, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ Models with D∞ = 10 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ Models with D∞ = 4 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ AS 7 6 5 4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 log (gcrit [cgs]) log ( ˙M [M⊙ yr−1]) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Mass-loss rate ˙M as a function of the gravitational acceleration at the critical radius gcrit = GMR−2 crit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Thin, dotted, gray lines indicate curves with ˙M ∝ g−3/2 crit .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' yielding log T2/3 K = (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='667 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='009) log ˙Mt M⊙ yr−1 + (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='03) (4) for the Sander & Vink (2020) sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (For comparison, the de- rived trend for the new sequences is shown as well in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=') We discuss possible origins later in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Quantitative mass-loss radius-dependence The similarity of the curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 5 and the simplicity of the slopes, indicates a common dependence between log ˙M and log Teff,crit for our sequences with fixed L/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Performing a linear fit, we obtain a relation in the form of log( ˙M [M⊙ yr−1]) = −6 · log(Teff,crit [K]) + offset (5) with the detailed fit coefficients being presented in appendix Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' A and Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The factor −6 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (5) implies that the obtained temperature dependence essentially reflects the radius change of the stellar models, since T 4 eff,crit ∝ R−2 crit and ˙M ∝ R3 crit (6) for models with L = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (as in our model sequences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Our corresponding plot (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1) further shows that deviations from the purely geometrical trend occur when we reach the limit of radiative driving discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 (and listed explicitly for each sequence in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1), as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' visible at the upper end of the WC sequence (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' From a dynamical perspective, the obtained Rcrit-trend for ˙M can be understood as a dependence on the gravitational acceler- ation on the critical point, where the wind is launched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' With the straight-forward definition of gcrit = g(Rcrit) = GM R2 crit (7) we can rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (6) as ˙M ∝ g−3/2 crit (8) since M is a constant among each of the model sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Trend curves reflecting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (8) are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 13 together with Article number, page 9 of 21 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' paper the curves from our model sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Generally, a decreasing trend of ˙M with log gcrit is not surprising as an increased grav- itational force needs to be overcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Given the content mass M along the model sequences, the change in gcrit expected from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (8) is purely geometrical, that is only from the change in Rcrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The model sequences align well with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (8), but there is a no- table flattening for the highest mass-loss rate, that is in the case of more dense winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' We cannot rule out that a numerical ef- fect is playing a role here as these high- ˙M models often operate on the limits of what the code is capable of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Nonetheless, given that the bending occurs in all sequences, a physical origin seems more likely and we continue our efforts on this assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For some sequences, there are also notable deviations from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (8) at the lower ˙M-end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' From the current set of calculations, the ap- parent kink in the curves approximately coincides with Te(Rrcrit) surpassing Teff,crit, meaning that the electron temperature at the critical point is higher than the effective temperature at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' However, the low number of models where this trend is clearly observed and the need to include higher ionization stages in these models, which can cause an additional offset in the numerical solutions if not done early enough in the sequence, currently re- frain us from concluding whether there is a clear “kink” or a more gradual change that might potentially be emphasized by a switch in the numerical setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In any case, the model solutions obtained in this thinner wind regime are characterized by (elec- tron) temperature stratification that remain very high, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' larger than 50 kK, until infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Their leading acceleration is provided by Fe M-shell ions, which are populated throughout the wind, qualitatively similar to the example shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 16 of Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' When considering the transformed mass-loss rate ˙Mt instead of ˙M, the scaling of the velocity with gcrit has to be considered as well, which we do in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Terminal velocity trends With the intrinsic solution of the hydrodynamic equation of mo- tion, our models automatically predict terminal wind velocities together with ˙M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' While already entering the transformed mass- loss rates, we now take a look at the explicit results for �∞ as a function of T2/3 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Although our sequences are not at all adjusted to match any particular observations, we also plot empirical results for WN stars obtained with PoWR for com- parison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' It is clear that our models match the general regime of the observed sample, but a closer inspection also shows caveats, for example with hydrogen-free WN stars showing values above the 20 M⊙ H-free sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Various possibilities could explain this (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', higher L/M and/or higher clumping), but a thorough investigation is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Impact of clumping In contrast to ˙M, the values for �∞ tend to scatter a bit more due to being evaluated at the outer boundary of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The ter- minal velocity further strongly depends on the included opacity, so including all ions contributing to the acceleration is necessary in order to avoid underestimating �∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' As apparent from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 14, �∞ also reacts on the choice of the clumping factor D∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' With a depth-dependent onset of the clumping, the response of the mass-loss rate to a change of D∞ is usually small as the result- ing differences in D(r) are small in the subsonic layers (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 1 in Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2020, and appendix Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' C of this work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In the supersonic layers, however, any differences in D∞ affect the bulk of the opacities being considered in the hydrodynamic equation of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Consequently, the obtained values for �∞ Empirical results MW H-free WN-s MW H-free WN-w LMC H-free WNs SMC WRs Model sequences 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0, Z⊙ 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ 15 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0, Z⊙ 20 M⊙, WC, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ Models with D∞ = 10 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ AS 1000 2000 3000 4000 5000 6000 7000 8000 20 60 100 140 180 220 T2/3 [kK] v∞ [km s−1] Hawcroft et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (ULLYSES) Vink & Sander (2021) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Terminal velocity as a function of T2/3 for the different model sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For comparison, also the derived trend for OB-star winds in the Milky Way (dashed line) and the LMC (dashed-dotted line) from Hawcroft et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=') are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' are notably higher for higher D∞, typically on the order of 20% when calculating a model for the same L, M, T∗, and chemical composition with D∞ = 50 instead of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In particular cases, the effects can be much larger, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' when a model has a deceleration regime in the case of a lower D∞, while there is no such regime for higher D∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Significant changes of the wind density regime due to a switch of D∞ can then lead to a stronger change in the derived ˙M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Moreover, the assumption of little to no clumping in the deeper layers would become invalid in case of a porous, optically thick medium, which can result from a subsonic, super- Eddington situation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Shaviv 1998, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Scaling with T2/3 Beside the differences due to clumping, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 14 demonstrates that any significant change of the chemical composition usu- ally affects the derived �∞-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In the figure, we see higher terminal velocities for the 20 M⊙ model sequence with surface hydrogen (XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2) compared to the corresponding hydrogen- free sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This result might be counter-intuitive at first as hydrogen does not provide significant line opacity that could be used to increase �∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' However, the additional hydrogen is able to boost the mass-loss rate of the star as the hydrogen atoms in the atmosphere provide a higher budget of free electrons compared to a hydrogen-free atmosphere1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' With more acceleration avail- able already in the deeper layers, the critical point of the wind moves inward to higher optical depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Line opacities which were subsonic in the hydrogen-free case can now be used to fur- ther boost the terminal wind speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Due to the higher mass loss of the hydrogen-containing model, the value of T2/3 decreases when comparing models with the same T∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Interestingly, as we saw in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1, the value of ˙Mt remains the same when com- paring against T2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In a follow-up study, we will test whether this behavior is universal when considering surface hydrogen or whether the chosen fraction of XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 coincidentally balances out other effects for a 20 M⊙ He-burning star as we e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' saw with the metallicity reduction being balanced by the surface hydrogen when considering only ˙M in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 1 A small hydrogen-layer on the surface has also the structural conse- quence of an increased stellar radius, which would again affect the wind parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Here, we discuss only the immediate atmospheric conse- quences for a fixed set of stellar parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Article number, page 10 of 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' : The temperature dependency of Wolf-Rayet-type mass loss To get some insights on the general scaling of WR-type winds with effective temperature (here: T2/3), we also compare our sequences to the trends for OB-type stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 14, we plot the trends obtained for the ULLYSES OB stars for the SMC and a Galactic comparison sample by Hawcroft et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=') as well as the predictions from Vink & Sander (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' We see that generally the terminal velocity increases much steeper with the temperature in OB-type winds than in WR-type winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Only at higher temperatures (T2/3 > 130 kK), when the winds become more optically thin as the mass-loss rates decrease, the steepness of the �∞(T2/3)-curves increases to a value more comparable to those obtained for OB-star winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Critical-point dependencies In addition to studying the behavior of �∞ as a function of the observable quantity T2/3, we further investigate the behavior of �∞ as a function of T∗ or the physically more meaningful Teff,crit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' As noted above, the values of �∞ tend to scatter a bit more, but if we restrict the linear fitting to the optically thick wind regime, we find log � �∞ [km s−1] � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 log (Teff(τcrit) [K]) + offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (9) Given that the number of models in the optically thick regime is restricted and not all sequences reach far enough to see the flattening of the trend, this coefficient has to be considered as rather uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Nonetheless, the corresponding scaling of �∞ ∝ R−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='6 crit can also be obtained from directly fitting �∞(Rcrit) in the optically thick limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Using the critical radius to define the escape velocity �esc = � 2GM Rcrit (10) we obtain �∞ ∝ �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 esc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (11) This relation also holds for the effective escape velocity �esc = � 2GM Rcrit (1 − Γe) (12) since all of the model sequences have a constant L/M and Γe is approximately constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The latter is consequence of the un- changed free electron budget below Rcrit in the considered tem- perature range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Thus, in the optically thick wind regime we have a slight difference with �∞ ∝ �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 esc,eff along the T∗-dimension com- pared to the well-known �∞ ∝ �esc,eff in the well-known CAK theory (named after Castor, Abbott, & Klein 1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For the se- quences along the L/M-dimension from Sander & Vink (2020), we instead obtain a negative trend of log �∞ ≈ −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='6 log �esc,eff + const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' in the optically thick regime with a flattening of the trend for the highest mass-loss rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In both cases, the scaling of �∞ with �esc,eff remains complicated with no straight-forward predic- tion as offsets remain in all scalings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This is in sharp contrast to the classical (m)CAK result, where �∞ follows as an offset-free value from �esc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For lower mass-loss rates (log ˙Mt < −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5), corresponding usually to Teff,crit > 150 kK, we reach the regime where winds are mostly optically thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Above, we could show that when reaching this regime, there seems to be an alignment of �∞(T2/3), with the slopes known from OB-type winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' With considerable scatter in the exponent of up to ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5, we find �∞ ∝ T 4 eff,crit, (13) Model sequences 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0, Z⊙ 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ 15 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0, Z⊙ 20 M⊙, WC, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ Models with D∞ = 10 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ Models with D∞ = 4 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ AS 6 5 4 3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 log (gcrit [cgs]) log ( ˙Mt [M⊙ yr−1]) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Transformed mass-loss rate ˙Mt as a function of the gravita- tional acceleration at the critical radius gcrit = GMR−2 crit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' To reflect the expected trends from the Rcrit-fits, thin gray lines are plotted in the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The dotted, gray lines indicate ˙M ∝ g−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 crit (optically thin regime), while the dashed, gray lines correspond to ˙M ∝ g−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='8 crit (optically thick regime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' corresponding to �∞ ∝ R−2 crit or �∞ ∝ �4 esc,eff, that is a steeper re- lation, contrary to the expected flattening of the slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' However, there is growing evidence from both observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Garcia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2014) as well as theoretical CMF-based and Monte Carlo calculations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Björklund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Vink & Sander 2021) that even in the typical OB-type regime the scaling of �∞ with �esc,eff is likely more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Influence on ˙Mt The scaling of �∞ with Rcrit introduces an additional dependency when considering the transformed mass-loss rate ˙Mt as a func- tion of Rcrit or gcrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (5) and (6), we know that ˙M ∝ T −6 eff,crit or ˙M ∝ R3 crit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' From the definition of ˙Mt (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2) we get log ˙Mt(Rcrit) = log ˙M(Rcrit) − log �∞(Rcrit) + offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (14) With the different trends derived for the optically thick and thin limit in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='3, we obtain ˙Mt ∝ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='6 crit and ˙Mt ∝ R5 crit respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Using gcrit as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (7) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (8), this yields log ˙Mt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='8 log gcrit + offset (15) for the optically thick limit and log ˙Mt = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 log gcrit + offset (16) in the optically thin limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' These trends are depicted as sets of gray lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 15, where the curves from the model sequences are shown as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In contrast to the ˙M(gcrit)-behavior discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='4, the representation of the slope in the optically thin regime is less precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In the optically thick limit, some curves align well, but others appear to be slightly steeper or shallower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Hence, the overall results for ˙Mt(gcrit) should be considered less robust than the ˙M(gcrit)-trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Interestingly, we do not see the “kink” or clear bending for some sequences at lowest (trans- formed) mass-loss rates that we see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 13 for ˙M(gcrit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' While it is hard to draw strong conclusions, there at least appears to be one continuous slope for ˙Mt(gcrit) in the thinner wind regime, regardless of whether the critical point is located at temperatures Article number, page 11 of 21 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' paper below or above Teff,crit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Since we find a change for ˙M alone, this would imply that �∞ outweighs this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' While the inspection of the corresponding sequences in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 14 is only indicative here, indeed the �∞-curves of these sequences bend again toward shal- lower slopes then plotting them as functions of Teff,crit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Potential consequences for stellar evolution Our study presents the very first sequences of hydrodynamically consistent atmosphere models in the cWR regime, where we vary the input parameter T∗ – corresponding roughly to Teff,crit for most models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' WR mass-loss recipes commonly do not incor- porate any temperature/radius dependency, which can be seen as a consequence of the optically dense winds of WR stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' When reproducing their spectra with prescribed velocity fields, there is a degeneracy of solutions making it impossible to find a unique value of T∗ for more dense winds (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Hillier 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Hamann & Gräfener 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Lefever et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' From an evolutionary standpoint, one could justify the omis- sion of a T∗-dependence arguing that hydrogen-free WN stars – and to some extend also WC stars – may form a 1D sequence as they represent He-burning stars that do not contain any further shell structure which could skew the relation between the lumi- nosity and mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In reality, effects such as inflation, convection, or rotation augment the physical conditions of wind launching and mass loss, especially when considering their multidimen- sional nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' However, in the currently typical 1D spherical ap- proach ignoring such issues, no further parameter would be nec- essary if the He star evolution could be perfectly mapped to one of the fundamental stellar parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For He stars above 10 M⊙, the intrinsic curvature of the HeZAMS indeed gets relatively small (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Langer 1989) and the obtained tracks of WR evo- lution in different codes yield very similar temperatures around log(T [K]) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='12, regardless whether these have been calcu- lated from pure He stars or including all prior evolution from the ZAMS (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Georgy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Limongi & Chieffi 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Higgins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Mass-loss comparison for a representative model In the models from Sander & Vink (2020), we thus ignored the width of ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1 dex in log T∗ and fixed T∗ in order to keep the total amount of models manageable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In this work, we now cal- culated a number of model sequences where we vary T∗ in order to investigate the effect of a wider range of T∗, which probes not only the curvature of the He ZAMS, but also gives a glimpse of how ˙M might be affected for stars which are not yet or no longer (exactly) on the He ZAMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' We find that despite the nar- row range in temperature, the effect on ˙M is quite noticeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For our 20 M⊙ model sequence (at Z⊙) even a narrow range of only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='05 dex (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', T∗ = 125 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 140) results in a factor of two in ˙M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Whether such a significant correction is really necessary de- pends on the difference between the most realistic choice of T∗ and the fixed value (141 kK) in Sander & Vink (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Combin- ing the structural constraints by Grassitelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2018) with our model sequence, we find an ideal value of Teff,crit ≈ T∗ ≈ 130 kK for a 20 M⊙ at Z⊙ model without any hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In Table 3, we provide a comparison of the resulting mass- loss rates for a 20 M⊙ star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Beside the values employing the new 2 In this discussion, we do not consider structure models that show hy- drostatic envelope inflation for more massive He stars (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 19 in Köhler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2015) as Grassitelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2018) demonstrated that such an inflation likely does not occur if a strong wind can be launched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Comparison of mass-loss rates obtained with different methods for a hydrogen-free WN star with log L/L⊙ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='7 and 20 M⊙ Paper log( ˙M [M⊙ yr−1]) Z⊙ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ Gräfener et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2017), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='-analytic(a) −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='72 no sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Gräfener et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2017), num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (b) −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='65 no sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Sander & Vink (2020) (D∞ = 50) −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='61 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='75 Sander & Vink (2020) (D∞ = 10) −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='64 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='83(c) this work, T∗ = 130 kK(d) (D∞ = 50) −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='40 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='56 this work, T∗ = 130 kK(d) (D∞ = 10) −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='42 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='83(e) Nugis & Lamers (2000) recipe −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='52 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='66 Hamann95+(f) recipe −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='40 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='65 Yoon (2017) recipe ( fWR = 1) −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='59 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='77 Yoon (2017) recipe ( fWR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='6) −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='39 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='57 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The ˙M determinations by Gräfener et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2017) employ the Prad- Pgas-plane with the sonic point conditions using (a) their Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (27) and assuming �∞ = 1800 km s−1 or (b) a numerically integrated dPrad/dr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (c) New calculation, but with T∗ = 141 kK as in Sander & Vink (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (d) Choice of T∗ based on matched Teff(τcrit) with Grassitelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2018) (e) Unstable solution close to driving breakdown, see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 (f) Mass-loss rates from Hamann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (1995) divided by a factor 10 and scaled with the Z-dependence from Vink & de Koter (2005), as suggested by Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' estimate of T∗ and the solutions for T∗ = 141 kK from Sander & Vink (2020), we also list the resulting ˙M-values from Gräfener et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2017) and commonly used (semi-)empirical recipes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For Z⊙ we find a difference of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 dex in ˙M, unaffected by the choice of D∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Using the values of Table 3 as an average mass- loss rate during the typical He burning lifetime (300 kyr), this corresponds to a difference between 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 M⊙ and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1 M⊙ at the end of core He-burning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This calculation is of course only a rough estimate and does not take any change of the stellar pa- rameters or surface abundances into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Nonetheless, the value using the ˙M from Sander & Vink (2020) is close to what we obtain with actual stellar evolution calculations in Higgins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' At 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙, a value roughly corresponding to the LMC, the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 dex shift holds as well when adopting D∞ = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For the hydrogen-free 130 kK model at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ with D∞ = 10, however, we only find a solution if we relax the stability criterion on ˙M between consecutive updates that we otherwise enforce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For the 141 kK model, we already see a notable difference in ˙M when reducing from D∞ = 50 down to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The reason is that we are already close to the regime where we can no longer obtain a wind solution driven by the hot iron bump (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For the 130 kK we have reached already a meta-stable situation with respect to the solution stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Thus, the obtained value of ˙M for D∞ = 10 is much lower than expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The matching of the ab- solute values for 130 kK and 141 kK is a pure coincidence with higher, also meta-stable solutions up to ≈ −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='7 for 130 kK being possible as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Structural limits and the role of hydrogen The presence of hydrogen at the surface can considerably change the limits of the wind onset derived above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In contrast to our hydrogen-free results shown in Table 3 and depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 6, our model sequence with XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ extends to much cooler temperatures (Teff,crit < 100 kK) as the additional accel- eration from free electrons helps to compensate the effect of the Article number, page 12 of 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' : The temperature dependency of Wolf-Rayet-type mass loss deceleration regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The choice of D∞ has some impact on the results, but in both cases the effect of surface hydrogen as such is much larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' While we do not aim at a detailed comparison with obser- vations in this work – which would require new analyses with dynamically-consistent models – it is striking that all hydrogen- free WN stars in the LMC are of the subtype WN4 or earlier (Hainich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Shenar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Moreover, all WC stars in the LMC show early subtypes as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' While one has to be careful drawing absolute conclusions, our temperature study now indicates that beside the metallicity limiting the lower lu- minosity of the observed WN population, the observed restric- tion of the subtype regime might be a direct consequence of the inability to launch WR-type winds below a certain (sonic point) temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' From the perspective of fixed stellar parameters, the lower mass-loss rate reached at a lower metallicity corresponds to a shift to earlier subtypes (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 4), thereby confirming the suggestion by Crowther et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Coming from a different angle, but addressing essentially the same problem, Grassitelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2018) and Ro (2019) used hydrodynamic stellar structure models and semi-analytic ap- proaches to predict the existence of a “minimum mass-loss rate” for launching a stellar wind from the hot iron bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For values of ˙M below this limit, extended low-density regions were predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In this work, we do not aim to obtain the latter type of solutions, but the calculations failing to launch a wind show a tendency toward trying to launch a wind further out with a (much) lower ˙M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' We can thus qualitatively confirm the structural predictions by Grassitelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2018) and Ro (2019), assuming that we are bound to a choice of T∗ following the – ideally hydrodynamical – structure calculations for the HeZAMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This underlines once more that unifying structural and atmosphere models remains a challenge that requires a new generation of both atmosphere and structure models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' An approximated handling of the temperature shift In light of the structural considerations above, it appears likely that any future description of WR-type mass loss needs a tem- perature or radius-dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Simpler treatments might be rea- sonable in a time-averaged situation, but cannot predict a real- istic mass loss for individual points along an evolutionary track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Hence, a detailed update of the Sander & Vink (2020) formula will eventually be necessary, but the current amount of models does not allow a wide-space parameter investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The appli- cability to lower masses also turns out to be nontrivial: One the one hand, the curvature of the HeZAMS toward cooler temper- atures should soften the sharp drop obtained in Sander & Vink (2020) of ˙M toward lower He star masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' On the other hand, we reach lower limits of Teff,crit for driving winds by the hot iron bump (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In our 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9 M⊙ sequence with XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, this limit is at ≈ 97 kK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Given that this limit seems to increase to slightly higher temperatures for lower L/M values, it appears un- likely that one can find any solution for a wind driven by the hot iron bump for stars with M ≤ 10 M⊙ fulfilling the L-M relation from Gräfener et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' While a full coverage of the driving limit at cooler tempera- tures will require its own tailored study, we can use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (5) from Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='4 to derive a decent temperature description up to dis- continuity in ˙M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (5) seems to be valid across both the optically thick and thin regime, we can approximate ˙M for WN winds driven by the hot iron bump via log � ˙M M⊙ yr−1 � = log � ˙MSV2020 M⊙ yr−1 � + 3 log � Rcrit Rcrit,T141 � (17) with ˙MSV2020 denoting the mass-loss rate from Sander & Vink (2020) and Rcrit,T141 = Rcrit(T∗ = 141 kK) being the critical ra- dius (in R⊙) of their corresponding model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='217 R⊙ for the 20 M⊙ He star without hydrogen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Although Rcrit,T141 could be obtained from L/M or Γe via a nonlinear fit of the Sander & Vink (2020) data, it is much more convenient to reformulate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (17) in terms of the effective temperature at the critical (≈ sonic) point Teff,crit, yielding log � ˙M M⊙ yr−1 � = log � ˙MSV2020 M⊙ yr−1 � − 6 log � Teff,crit 141 kK � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (18) Apart from small deviations for the highest mass-loss rates ( ˙M ≫ 10−4 M⊙ yr−1), the fixed value of 141 kK accurately repre- sents the value of Teff,crit in Sander & Vink (2020), as illustrated previously in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This adjusted ˙M-recipe requires the knowledge of either Teff,crit or Rcrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' As we did include only a small microturbulent ve- locity in our modeling efforts (30 km s−1), the quantities can be replaced by the sonic point values without introducing a consid- erable error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Still, the accurate use of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (17) and (18) requires models with a meaningful sonic point in a hydrodynamical sense to prevent reintroducing any further radius/temperature discrep- ancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Stellar atmosphere analyses typically employ predefined velocity fields (usually β-laws) and thus do not have a sonic point that is hydrodynamically consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Purely hydrostatic stellar structure calculations are problematic as well as they do not yield a sonic point by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This underlines that in order to ob- tain a really insight- and meaningful comparison between theory and observation for optically thick winds, a new generation of both atmosphere and stellar structure models will be necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The results obtained in our study could be helpful to even- tually obtain realistic predictions for the effective temperatures (T2/3) of WR stars in stellar evolution models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' A route toward such a recipe based on our findings is given in appendix Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Transparency to He II ionizing photons Despite having intrinsically quite hot temperatures, classical WR stars do not necessarily emit a significant number of ionizing photons beyond the He ii ionization edge, that is below 227 Å or above 54 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' As first described in Schmutz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (1992), the transparency of the wind for photons with energies above 54 eV depends on the mass-loss rate, and thus the density of the wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In more dense winds, He iii recombines to He ii, making the at- mosphere opaque to He ii ionizing photons out to very large radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The presence of line blanketing further affects the absolute ion- izing fluxes significantly (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Beside usually leading to a reduction of the He i ionizing flux, it can also affect the He ii ionizing flux transition by a few orders of magnitude as we will see in our model calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' To cover the region where the (continuum) optical depth drops below unity in this wavelength region, we extended the outer boundary radius Rmax of our atmosphere models to extremely large values, often up to 100 000 R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (Typical atmosphere models for spectral fitting re- quire only Rmax = 100 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 1000 R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=') In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 16, we plot the rate of ionizing photons per second QHe ii as a function of the transformed mass-loss rate ˙Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The ab- solute numbers in the regime with low QHe ii are more uncertain Article number, page 13 of 21 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' paper Model sequences 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0, Z⊙ 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ 15 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0, Z⊙ 20 M⊙, WC, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ Models with D∞ = 10 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ Models with D∞ = 4 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ AS 38 41 44 47 50 5 4 3 log( ˙Mt [M⊙ yr−1]) log(QHeii [s−1]) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Number of helium ionizing photons per second QHe ii as a func- tion of the transformed mass-loss rate ˙Mt for the new model sequences calculated in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 log ( ˙Mt [M⊙ yr−1]) 38 40 42 44 46 48 50 log QHe ii 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 Z⊙ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 Z⊙ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 Z⊙ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1 Z⊙ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='05 Z⊙ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='02 Z⊙ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Number of helium ionizing photons per second QHe ii as a func- tion of the transformed mass-loss rate ˙Mt for the model sequences com- puted in Sander & Vink (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' as they depend strongly on the precise boundary treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' If the wind becomes transparent around and below 227 Å only very close to the model boundary or even remains optically thick at Rmax, the value for QHe ii can be underestimated, but should never exceed 1041 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Given that this is many orders of magnitude be- low the actually strong QHe ii emitters with rates > 1047 s−1, the values of the shaded regime in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 16 should have no practical consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Displaying the He ii ionizing flux as a function of ˙Mt in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 16 confirms that similar to other quantities, the switch in transparency is caused by the lower wind density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In fact, there seems to be a critical lower boundary around log( ˙Mt [M⊙ yr−1]) = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='6 to −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='4 where all model sequences switch abruptly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' To study whether this transition value might be more universal, we created the same plot for the model se- quences from Sander & Vink (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Their model set, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 17, clearly hints at a Z-dependency for the transition, which we do only sparsely map in our new model set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In fact, our new sequences are quite complementary to the datasets from Sander & Vink (2020), indicating that the transition does not only de- pend on Z as a total value, but likely on the detailed composi- tion and – notably especially at the lower end of the transition in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 16 – on the choice of the clumping factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Nonetheless, we can conclude that all stars in our large model sample with log( ˙Mt [M⊙ yr−1]) < −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='6 are strong emitters of He ii ionizing flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Thus, we propose to take this value as an upper limit of whether to consider WR stars as notable contributors to the He ii ionizing photon budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Summary and conclusions In this work, we presented an exploratory study for the temperature-dependency of radiation-driven winds launched by the so-called hot iron opacity bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For the first time, we cal- culated temperature-dependent sequences of hydrodynamically consistent stellar atmosphere models in the cWR regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' To achieve our results, we had to allow for nonmonotonic velocity field solutions when solving the hydrodynamic equation of mo- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In order to perform the necessary radiative transfer in the comoving frame, we afterwards interpolated the obtained veloc- ity fields such that the main wind properties ( ˙M, �∞) as well as the characteristics in the outer wind were maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' We draw the following conclusions: – The mass-loss rates ˙M depend significantly on the critical radius Rcrit and thus also on the assumed model temperature setting Teff(Rcrit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For model sequences with constant lumi- nosity L and stellar mass M, we obtain ˙M ∝ R3 crit over a wide range with moderate deviations from this purely geo- metrical effect occurring at the lower and upper end of our sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This finding can also be expressed in the form of ˙M ∝ g−3/2 crit , reflecting that larger radii for the critical point imply a lower gravitational force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Our findings underline that WR-type mass-loss depends on multiple parameters and the 2D description from Sander & Vink (2020) needs to be ex- tended further to describe all relevant effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' – Except for very dense winds – corresponding to log( ˙Mt [M⊙ yr−1]) ≈ −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 and above – the effective temperature at the critical point Teff(τcrit) is close to the effective temperature at a Rosseland continuum optical depth of τR,c = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For WN-type models τR,c = 20 typically corresponds to τThom ≈ 17, albeit with considerable scatter along the model sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' – We find a characteristic value of log( ˙Mt [M⊙ yr−1]) ≈ −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 for the transition between the optically thin and thick regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' While there is some scatter between different model se- quences, this characteristic value of ˙Mt (plus some error mar- gin) provides a very convenient tool to distinguish between the regimes as ˙Mt can also be determined with empirical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Known WC stars show values well above this (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Gräfener & Vink 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2019) while WO stars might be found on both sides of the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Whether the characteristic value of ˙Mt also holds for winds that might not be driven by the hot iron bump is currently unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' We cal- culated the transformed mass-loss rates for stars at the spec- tral transition from Of to WNh, which likely happens at a cooler temperature regime than studied in this work3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Their corresponding transformed mass-loss rates ˙Mt,trans appear to be below −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' at log( ˙Mt,trans [M⊙ yr−1]) ≈ −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 in the Arches cluster (Martins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Vink & Gräfener 2012) 3 The transformed mass-loss rate ˙Mt as such should not be confused with the transition mass-loss rate ˙Mtrans from Vink & Gräfener (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Nonetheless, if the other necessary parameters are known, one can es- timate the corresponding transformed mass-loss rates for stars defining the transition mass-loss rate, denoted as ˙Mt,trans Article number, page 14 of 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' : The temperature dependency of Wolf-Rayet-type mass loss and even lower in R136 (Bestenlehner 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Bestenlehner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' – The choice of the maximum clumping factor D∞ does not affect our derived ˙M(T2/3) trends, but leads to an additional shift in the obtained relations with higher clumping factors corresponding to higher ˙M for the same T2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In contrast to all shifts introduced by varying fundamental stellar parame- ters or abundances, the shift due to D∞ does not vanish when considering ˙Mt(T2/3) instead of ˙M(T2/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' – In the limit of optically thick winds, we obtain a linear re- lation between log T2/3 and log ˙Mt, independent of chem- ical composition (but for a fixed clumping factor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Com- bined with the also Z-independent result from Sander & Vink (2020) that log ˙Mt ∝ log(L/M), this could provide an easy- to-use prediction for WR effective temperatures in stellar structure and evolution models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' – Classical WR stars and non-WR helium stars with log( ˙Mt [M⊙ yr−1]) < −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='6 are strong emitters of He ii ion- izing flux (with QHe ii > 1048 s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Helium stars with stronger winds are (mostly) opaque to radiation above 54 eV and thus should not be considered as sources of hard ionizing radia- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' – Albeit being limited in comparability to particular observed targets, our findings indicate that high clumping factors (D ≈ 50) might be necessary to reproduce the observed combina- tions of ˙M and �∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This is in sharp contrast to the first results obtained from 3D wind modeling by Moens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2022) ar- guing for much lower clumping factors of D ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' At present, the reason for this discrepancy is unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Various solutions are possible, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', missing opacities in our wind models – where the presently assumed high clumping would act as a “fudge factor” to make up for that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Alternatively, the sharp contrast in the clumping factor might simply be the result of a mismatch between the considered regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Currently, the 3D models from Moens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2022) probe only the wind onset where even in our 1D models we assume D(r) ≪ D∞ with D(τcrit) typically ranging between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' – When comparing empirically obtained results in the T2/3- ˙Mt- plane to our derived curves, we find a significant fraction of stars to have lower values of ˙Mt than predicted by our curves using hydrodynamic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' It is currently unclear whether this is due to a deviation from the theoretical setup in this work (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', different clumping stratification, other L-M com- binations) or inherent simplifications in the empirical analy- ses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', the use of a β-type velocity law).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' A dedicated anal- ysis of individual objects with hydrodynamical model atmo- spheres will be necessary to uncover the origin of this dis- crepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' – The limits of driving optically thick winds crucially de- pend on our knowledge of opacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In case of consider- able changes – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', a higher iron opacity as reported by Bailey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2015) for the so-called “deep iron bump” at Te ≈ 2 · 106 K – wind quantity predictions such as ˙M and �∞ could shift significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Moreover, our understanding of the limits of radiative driving would be affected as well, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', due to strengthening or weakening the bumpy radius dependency of the flux-weighted mean opacity κF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Beside the impact of multi-D effects, higher (Fe) opacities could play an impor- tant role to resolve current discrepancies, such as the lower luminosity end of the LMC WN population or the aforemen- tioned need for higher D∞ to reach the observed terminal velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' With these conclusions, our study underlines the complexity of radiation-driven mass loss, revealing both parameter regimes with a clear scalings and characteristic transitions as well as more obscure parameter regions where ˙M appears to break down suddenly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' We provide an adjustment of the recent ˙M-description from Sander & Vink (2020) to account for different radii (or ef- fective temperatures respectively) and emphasize that the model efforts presented there as well as in this work were limited to the regime where winds are launched by the hot iron bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' We thus consider our work as an intermediate step on the way toward a more comprehensive understanding of WR-type mass loss and will expand to other regimes in future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The authors would like to thank the anonymous referee for their careful and constructive comments and suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' AACS and VR ac- knowledge support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) in the form of an Emmy Noether Research Group – Project-ID 445674056 (SA4064/1-1, PI Sander).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' RRL is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 138713538 – SFB 881 (“The Milky Way System”, subproject P04).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' LP acknowl- edges support by the Deutsche Forschungsgemeinschaft – Project-ID 496854903 (SA 4046/2-1, PI Sander).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' JSV is 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2022, MNRAS, 513, 5606 Nakauchi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' & Saio, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2018, ApJ, 852, 126 Nugis, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' & Lamers, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Bruzual, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2019, MNRAS, 490, 978 Poniatowski, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Sundqvist, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', 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P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2018, A&A, 610, A60 Sander, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Hamann, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2020, MNRAS, 499, 873 Sander, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Vink, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', & Hamann, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2020, MNRAS, 491, 4406 Schaerer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Contini, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', & Kunth, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 1999, A&A, 341, 399 Schmutz, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Leitherer, C.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' & de Koter, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2005, A&A, 442, 587 Vink, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' & Gräfener, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2012, ApJ, 751, L34 Vink, J.' metadata={'source': 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Woosley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Sukhbold, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', & Janka, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2020, ApJ, 896, 56 Yoon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2017, MNRAS, 470, 3970 Yoon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Langer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', & Norman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2006, A&A, 460, 199 Yusof, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Hirschi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Eggenberger, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2022, MNRAS, 511, 2814 Article number, page 16 of 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' : The temperature dependency of Wolf-Rayet-type mass loss AS 6 5 4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='4 log(Teff(τcrit) [K]) log( ˙M [M⊙ yr−1]) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Linear fits (solid lines) to the Mass-loss rate ˙M as a function of Teff(τcrit) in a double-logarithmic-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The fit coefficients for the different datasets are given in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Appendix A: ˙M-temperature Fit For all of our model sequences, the data points in the log ˙M- log Teff(τcrit)-plane suggest a linear relation between the two quantities with deviations occurring only close to the wind driv- ing limit (corresponding to the maximum ˙M in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In the fits, we thus exclude the uppermost 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='15 dex in log ˙M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The result- ing fit coefficients for the slope including their error margins are given in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For each individual sequence, the mass loss can be well described for Teff(τcrit) > Teff,crit,min and ˙M < ˙Mmax by log � ˙M M⊙ yr−1 � = −6 log � Teff,crit 141 kK � + log � ˙Moffset M⊙ yr−1 � (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1) with the value for ˙Moffset being different for each model se- quence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1 also lists these coefficients together with the corresponding validity limits Teff,crit,min and ˙Mmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Appendix B: T2/3 temperatures for the Sander & Vink (2020) sample The effective temperatures T2/3 at τRoss = 2/3 resulting from the model sequences in Sander & Vink (2020) are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Similar to what we obtain when varying T∗, cooler values of T2/3 require higher mass-loss rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' As T∗ is fixed to ≈ 141 kK in Sander & Vink (2020), higher luminosities or L/M- ratios are required to reach higher mass-loss rates for the same Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' At lower metallicity, stars have to get closer to the Edding- ton Limit to reach sufficient mass loss, shifting the onset of the drop in T2/3 to higher L and steepening in particular this drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The differences in T2/3 and the range of luminosities covered has quite some interesting implications on the spectral appearance of the stars and consequently also which WR subtypes one would expect in a certain environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Assuming that at least all the winds of early-type WR stars are launched at the hot iron bump, the temperatures of the lowest luminosity WN stars should get hotter at low Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This seems to be the case when comparing the WN populations in the Milky Way and the LMC (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Hamann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Shenar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2020), but further – ideally hydrogen- free – WN populations in other Galaxies need to be studied to draw any firm conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In the SMC, apart from the small sample size, all WN stars contain hydrogen and might not align 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 log (T2/3 [kK]) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 log (L [L⊙]) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 Z⊙ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 Z⊙ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 Z⊙ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1 Z⊙ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='05 Z⊙ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='02 Z⊙ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' HRD with the effective temperature T2/3 defined at a Rosse- land optical depth of τRoss = 2/3 and the model luminosity L for our sets of He ZAMS models at different Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For comparison, the HeZAMS (gray, dashed) and ZAMS (gray, solid) are shown as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' with the L-M relation we assume in Sander & Vink (2020) and this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Appendix C: The effect of enhanced clumping on the radiative acceleration The effect of clumping on our hydrodynamic wind solutions is not trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Given that we only use the so-called microclumping approximation assuming optically thin clumps and the solution of radiative transfer in the comoving frame is performed with the average density and not the clumped density, one might ex- pect that the choice of D∞ could have no effect at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' However, this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Different values of D∞ affect the population numbers which in turn affect the radiative transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In particular, higher choices of D∞ favor recombination in the wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For most elements, lower ionization stages provide more opacity and thus more radiative acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 we present the resulting acceleration contributions for the hydrogen-free 20 M⊙ WN models with D∞ = 50 and 10, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' To obtain these curves, the in- dividual opacities resulting from the different ions are stored in addition to the total opacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Beside the total calculation of arad(r) = 4π c ∞ � 0 κν(r) Hν(r) dν (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1) similar integrals to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1) are calculated using only the ion- specific opacities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' κFe V ν ) instead of the total κν, yielding the specific acceleration contribution for each ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The correspond- ing wind parameters of the two displayed models and two other sets with varying D∞ are listed in Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' With out fixed char- acteristic velocity for the clumping onset of �cl = 100 km s−1, the depicted models increase from almost no clumping to D∞ within the hot iron bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Thus, the mass-loss rates are barely affected, but the terminal velocity increases significantly from 1186 km s−1 to 1754 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The comparison of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1 confirms that the additional opacity to reach the higher terminal velocity is pro- vided by lower ions, most notably Fe iv, which is the leading accelerator in the outer wind for the model with D∞ = 50, while Fe v remains in the lead for the model with D∞ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In both Article number, page 17 of 21 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' paper Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Linear fit results for log ˙M versus log Teff,crit plus offsets and limitations for the temperature-dependent mass loss of our model sequences described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Sequence slope formal log ( ˙Moffset [M⊙ yr−1]) Teff,crit,min [kK] log ( ˙Mmax [M⊙ yr−1]) M [M⊙] XH Z [Z⊙] D∞ error WN 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 50 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='03 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='33 92 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='60 WN 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 50 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='05 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='31 88 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='51 WN 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 50 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='06 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='09 94(br) −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='20 WN 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 50 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='07 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='92 105(br) −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='36 WC 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 50 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='05 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='81 118(br) −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='60 WN 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 50 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='04 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='13 98 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='62 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (br) For marked sequences, Teff,crit,min reflects the lower limit for winds driven by the hot iron bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In all other sequences breakdown, this values just refers to the minimum explored value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Derived wind parameters for WN models with different D∞ D∞ log ( ˙M [M⊙ yr−1]) �∞ [km s−1]) log ( ˙Mt [M⊙ yr−1]) WN, 20 M⊙, XH = 0, Z⊙ 10 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='42 1186 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='77 50 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='40 1754 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='57 WN, 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ 4 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='34 1194 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='89 10 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='28 1448 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='71 50 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='28 1970 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='50 WN, 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ 4 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='44 609 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='70 10 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='35 848 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='56 50 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='28 1394 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='35 cases Fe v is the most populated Fe ion in the outermost wind, while the “fresh” opacity provided by the lesser populated Fe iv is most efficient for the line acceleration in the case of D∞ = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In the case of D∞ = 10, the population of Fe iv instead is too low to contribute significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The change in �∞ is further enlarged by the significantly smaller deceleration zone in the D∞ = 50 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In the deeper wind layers, the higher clumping boosts the contribution from the iron M-shell opacities and leads to an in- creased bound-free contribution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' recombination) from He ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The two other examples in Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1 illustrate that in some cases also the mass-loss rate can be notably affected by changes of D∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In our models, this is a consequence of the fixed value of �cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For regimes where generally lower values of �∞ are reached, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' in lower metallicity model set, often the whole amount of acceleration is reduced, shifting also the region with � ≈ 100 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This can then have two effects leading to a lower ˙M for lower values of D∞: first, a direct reduction of opacities in the region that determines ˙M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In addition, the reduced wind den- sity could push the star out of the regime where the critical point is in a totally optically thick region (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Sander & Vink 2020), which would lead to a further reduction in ˙M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In the last column of Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1, we provide the resulting transformed mass-loss rates ˙Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' While there is already a scal- ing with ˙M √D∞ in these, it does not compensate the clumping changes as the square root of the D∞-ratios is much larger than the changes in �∞ (and ˙M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For example, the hydrogen free mod- els differ by √50/10 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='24, while �∞ only increases by a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Therefore, the models with higher D∞ posses a higher ˙Mt, despite larger terminal velocities reducing its value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Appendix D: Estimating effective temperatures in stellar structure models For stellar structure models, the occurrence of optically thick winds usually spoils the straight-forward prediction of the ob- servable effective temperature T2/3 (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Groh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2014, for a more detailed discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In the previous Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='3, we ob- tained that for a given clumping factor D∞, our model sequences collapse almost perfectly to a single line in the ˙Mt-T2/3-plane, yielding T2/3 ∝ ˙M−1/2 t (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1) for log ( ˙Mt [M⊙ yr−1]) > −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5, thereby providing us with a potential path to predict the observable effective temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The relation (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1) seems to be approximately unaffected by abundance (Xi) changes, but there is a clear offset for differ- ent choices of D∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In our calculations, there is a difference of ∆ log (T2/3 [K]) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='08 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1 between D∞ = 4 and D∞ = 10 and ∆ log (T2/3 [K]) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='15 between D∞ = 10 and D∞ = 50, but the current amount of data along the D∞ plane is insufficient to provide a robust mathematical formula that could enable a scaling with D∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' A different slope of ≈ −2/3 was obtained in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='3, when considering the sample of Sander & Vink (2020) instead of our new model sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The origin of the difference in the slopes must be rooted in the different nature of the sequences: In the new sequences calculated for this work, L and M are constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For higher mass-loss rates ˙M we then obtain lower values of �∞ along a sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In the sequences from Sander & Vink (2020), where we proceed to higher L/M-ratios along each dataset, such a trend between ˙M and �∞ is only reached in the optically thin part, while we obtained ˙M ∝ �∞ in the dense wind regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' As a consequence, models from the two different sources with ap- proximately the same value of ˙M will differ in their �∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' When comparing the �∞-values, the models from the T∗-sequences in this work will have lower terminal velocities and thus their ˙Mt will be higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Arguing that ˙M is the major factor in setting the T2/3 value, we can thus conclude that this difference in the �∞ trends leads to the steeper slope for the sequences along the L/M-domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Given the focus of this work on the temperature trends and the fact that the steep linear trends in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 12 do not provide a good description around the transition region of log ( ˙Mt [M⊙ yr−1]) ≈ −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5, we therefore suggest the more shal- low formula log (T2/3 [K]) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 log ( ˙Mt [M⊙ yr−1]) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2) as a first attempt to approximate the observable effective tem- perature T2/3 of a WR star in stellar structure models, which is Article number, page 18 of 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' : The temperature dependency of Wolf-Rayet-type mass loss ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='Fe X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='Fe XI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='Fe XII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='Fe XIII ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='Fe XIV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='Fe XV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='Fe XVI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='arad ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='apress ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='aThom ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='AS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 3 2 1 0 1 2 3 4 log (r/R∗ − 1) log (a/g) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Contributions of the different ions to the radiative acceleration of a hydrodynamically consistent, hydrogen-free WN model with T∗ = 130 kK, log L/L⊙ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='7, M = 20 M⊙, and D∞ = 50: Different ions are denoted by a combination of different color and symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The total radiative acceleration (arad), the Thomson acceleration from free electrons (aThom = Γe · g), and the contribution from gas (and turbulence) pressure (apress) are also shown for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The loosely dashed horizontal line denotes the total Eddington limit that needs to be overcome to launch a wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' essentially a rounded version of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This formula implicitly assumes D∞ = 50 and is recommended for ˙Mt > 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 M⊙ yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For lower estimates of D∞, the offset value of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9 would need to be reduced by about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' We emphasize that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2) is a first approach that needs to be tested and likely refined in future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' With Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2) given, only the transformed mass-loss rate ˙Mt needs to be known to determine T2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In Sander & Vink (2020), we could show that for T∗ = 140 kK the quantity ˙Mt is practi- cally independent of metallicity in the limit of optically thick winds (“pure WR regime”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' There, ˙Mt can be expressed as a linear function of L/M with a possible deviation only occur- ring for He stars with current masses above 50 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' To check whether this conclusion is independent of T∗, we calculate a small set of models sequences with different L/M values for dif- ferent Teff,crit ≈ T∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The resulting trends are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' It is clear from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1 that there is some uncertainty in the slopes as well as a potential dependence of the slopes on T∗ itself, but in general an approximately linear behavior is obtained for each choice of Teff,crit ≈ T∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Hence, we obtain a viable prediction method for stellar evolution models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This method is particularly elegant as it does not require any further assumptions about the flux-weighted mean opacity or the shape of the velocity field as for example necessary in the current wind-corrected tempera- tures in the GENEC models (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', Groh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' To get a formula for ˙Mt that only depends on quantities which can be obtained from stellar structure calculations, we can use the result derived in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Considering that in the new model sequences calculated for this work both L and D∞ are constant within one sequence, we can conclude that ˙Mt ∝ ˙M/�∞ and obtain log ( ˙Mt [M⊙ yr−1]) = −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 · log (Teff(τcrit) [K]) + offset (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='3) or ˙Mt ∝ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='6 crit in the regime of optically thick winds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' for log ( ˙Mt [M⊙ yr−1]) > −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In a second step, we then merge Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='3), which has been determined for sequences of constant L/M, with the L/M- dependence obtained in Sander & Vink (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Together, we synthesize the formula log ˙Mt M⊙ yr−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='25 log L/M L⊙/M⊙ + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='6 log Rcrit R⊙ + ˙Mt,off(Xi, D∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='4) Again, it might be more convenient to replace Rcrit with Teff,crit and gauge this with the 20 M⊙ model at 141 kK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This then yields Article number, page 19 of 21 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' paper He I He II He III C III C IV N III N IV N V O III O IV O V Ne III Ne IV Ne V Ne VI Ne VII Ne VIII Na III Na IV Na V Na VI Mg V Mg VI Si IV P V S IV S V S VI Cl IV Cl V Cl VI Ar IV Ar V Ar VI Ar VII K IV K V K VI Ca III Ca IV Ca V Ca VI Ca VII Fe IV Fe V Fe VI Fe VII Fe VIII Fe IX Fe X Fe XI Fe XII Fe XIII Fe XIV Fe XV Fe XVI arad apress aThom AS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 3 2 1 0 1 2 3 4 log (r/R∗ − 1) log (a/g) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Contributions of different ions to the radiative acceleration, plotted similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1, but now for a model employing D∞ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The wind reaches a lower terminal velocity and the supersonic region with Γrad = arad/g < 1 is more pronounced than in the model with D∞ = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' AS 5 4 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='6 log(L/M [L⊙/M⊙]) log( ˙Mt [M⊙ yr−1]) M∗ [M⊙] 13 16 20 30 50 T∗ = 100 kK T∗ = 110 kK T∗ = 120 kK T∗ = 130 kK T∗ = 141 kK T∗ = 150 kK T∗ = 160 kK Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Transformed mass-loss rate ˙Mt as a function of L/M for dif- ferent sequences varying in T∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' All models use XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 except the dashed-dotted sequences having XH = 0 for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Apart from the red, dashed sequence (using D∞ = 10), all sequences employ D∞ = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The gray, dotted line is an interpolation of the solid sequences along the theoretical temperatures for the He ZAMS from Grassitelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (2018) and Langer (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The light dotted linear curves in the background indicate the slope of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='25 which is used in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='4) and onward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' log ˙Mt M⊙ yr−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='25 log L/M L⊙/M⊙ − 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 log Teff,crit 141 kK − 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5), we also dropped the offet ˙Mt,off(Xi, D∞) which con- tains further, uncertain dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' These can alter the re- sulting values of ˙Mt, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' by ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 dex when changing from D∞ = 50 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Inserting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2), we obtain the final formula for estimating T2/3: log �T2/3 K � = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='595 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='625 log L/M L⊙/M⊙ + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='6 log Teff,crit 141 kK, (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='6) This formula is only valid for hydrogen-free WN stars as we have considerable offsets for other chemical compositions in ˙Mt(L/M) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' While the conversion between ˙Mt and T2/3 is unaffected by chemical composition (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='3), the re- sulting radiative acceleration is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For example, the additional acceleration from free electrons in partially stripped stars with remaining surface hydrogen leads to higher mass-loss rates than in H-free stars of the same L/M-ratio (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 5), thereby sub- stantially shifting the balance between ˙M and �∞ and the result- ing ˙Mt-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In a future study, we thus plan to extend Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='6) by a hydrogen-dependent term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' While various uncertainties, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=', of the precise slopes in ˙Mt(L/M) and T2/3( ˙Mt) limit the accu- racy of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='6), it is sufficient enough to tell whether observed Article number, page 20 of 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' : The temperature dependency of Wolf-Rayet-type mass loss Empirical results MW H-free WN-s MW H-free WN-w LMC H-free WNs SMC WRs Model sequences 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0, Z⊙ 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ 15 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0, Z⊙ 20 M⊙, WC, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ Models with D∞ = 10 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ AS 5 4 20 60 100 140 180 220 T∗ [kK] log( ˙M [M⊙ yr−1]) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Analogous plot to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 5, but now showing the mass-loss rate ˙M as a function of T∗ for our model sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Model sequences 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0, Z⊙ 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ 20 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, Z⊙ 15 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='0, Z⊙ 20 M⊙, WC, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ AS 6 5 4 3 20 60 100 140 180 220 T∗ [kK] log( ˙Mt [M⊙ yr−1]) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Analogous plot to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1, but now showing the transformed mass-loss rate ˙Mt instead of the normal ˙M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' effective temperatures of predicted objects are in the range of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 100, 50, or only 20 kK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Depending on the scientific con- text, such differences in T2/3 can have a big impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' While we do not have enough data to draw larger conclusions, the thick gray dotted line Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1 illustrates that on the He ZAMS, com- pact radii of the stars at the critical point might even outweigh an expected increase in ˙Mt due to a larger L/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Nonetheless, we can present the first estimate of T2/3 for WN stars derived from fundamental principles which can be readily applied in stellar evolution models and population synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The validity of the assumptions made here will have to be tested within dedicated test calculations in stellar evolution models and benchmarked with WR observations analyzed with traditional as well as dy- namically consistent models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Appendix E: Additional Figures In addition to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 5 discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 3, we plot the mass-loss rate ˙M as a function of T∗ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' For very high mass-loss rates, we see a bending of the curves toward lower values of T∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' This is a consequence of the deeper wind launching, which in this regime happens further in than the defining optical depth for T∗ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' at τR,cont > 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Numerically, these models use a higher optical depth as their inner boundary and we then deter- Teff(τR = 2/3) T∗(τR,c = 20) Teff(τcrit) Te(Rcrit) AS 6 5 4 20 60 100 140 180 220 T [kK] log( ˙M [M⊙ yr−1]) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Mass-loss rates as a function of different temperature scales for a series of dynamically consistent atmosphere models with log L/L⊙ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='35, M = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9 M⊙, and XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2: The thick red dashed line denoted the classical effective temperatures defined at a Rosseland optical depth of τR = 2/3, while the green solid line and the blue dashed-dotted lines denotes the effective temperatures referring to τcrit and τR,cont = 20 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' The green dotted line on the right denotes the (electron) temperature at the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Curves in lighter colors reflect mod- els using the simple integration treatment suppressing negative velocity gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Teff(τR = 2/3) T∗(τR,c = 20) Teff(τcrit) Te(Rcrit) AS 6 5 20 60 100 140 180 220 T [kK] log( ˙M [M⊙ yr−1]) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Analogous plot to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='3, but now for the WC model series with log L/L⊙ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='7, M = 20 M⊙, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' mine T∗(τR,cont = 20) for a better comparison with the rest of the model calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In this regime, T∗ is no longer a good approximation for Teff,crit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' To eliminate the effect of different clumping factors and any remaining distance uncertainties, it is helpful to consider the transformed mass-loss rate ˙Mt instead of ˙M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, we show the analogous plot to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='1 with ˙M being replaced by ˙Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' While the vertical spread in the observations is slightly re- duced, the general temperature mismatch between the empirical T∗ and our model sequences remains, highlighting once more the “Wolf-Rayet radius problem” seen in traditional atmosphere analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' In an extend to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 6 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 7, we show similar plots for the model sequences with 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='9 M⊙, XH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2 and Z = Z⊙ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='3 and the WC model sequence with 20 M⊙ and Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='5 Z⊙ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Similar to the result obtained for the 15 M⊙-sequence discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content='2, there is a lower minimum temperature for obtaining wind solutions driven by the hot iron bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} +page_content=' Article number, page 21 of 21' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAzT4oBgHgl3EQf0f6b/content/2301.01785v1.pdf'} diff --git a/TdE2T4oBgHgl3EQftAiy/content/2301.04066v1.pdf b/TdE2T4oBgHgl3EQftAiy/content/2301.04066v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..1643215aaa5c3ed7604d33932de712353fc2a845 --- /dev/null +++ b/TdE2T4oBgHgl3EQftAiy/content/2301.04066v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2800ba1f69fa2305e89b96f3317ac2893ab5e9d3ab020cf5cada976469761f83 +size 5742593 diff --git a/UNE2T4oBgHgl3EQfCwYa/content/tmp_files/2301.03616v1.pdf.txt b/UNE2T4oBgHgl3EQfCwYa/content/tmp_files/2301.03616v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a72263aa0b405ea77de9c78fe95acac1d26e1499 --- /dev/null +++ b/UNE2T4oBgHgl3EQfCwYa/content/tmp_files/2301.03616v1.pdf.txt @@ -0,0 +1,2241 @@ +Power Corrections to Energy Flow Correlations from Large Spin Perturbation +Hao Chen,1, ∗ Xinan Zhou,2, † and Hua Xing Zhu1, ‡ +1Zhejiang Institute of Modern Physics, School of Physics, Zhejiang University, Hangzhou, 310027, China +2Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China. +Dynamics of high energy scattering in Quantum Chromodynamics (QCD) are primarily probed +through detector energy flow correlations. One important example is the Energy-Energy Corre- +lator (EEC), whose back-to-back limit probes correlations of QCD on the lightcone and can be +described by a transverse-momentum dependent factorization formula in the leading power approxi- +mation. In this work, we develop a systematic method to go beyond this approximation. We identify +the origin of logarithmically enhanced contributions in the back-to-back limit as the exchange of +operators with low twists and large spins in the local operator product expansion. Using techniques +from the conformal bootstrap, the large logarithms beyond leading power can be resummed to all +orders in the perturbative coupling. As an illustration of this method, we perform an all-order re- +summation of the leading and next-to-leading logarithms beyond the leading power in N = 4 Super +Yang-Mills theory. +INTRODUCTION +Distributions of energy flows in high energy scattering +in Quantum Chromodynamics (QCD) encode unique in- +formation of the Lorentizian dynamics of quantum gauge +theory, which is otherwise hard to extract. +A famous +example is the formation of jets, which are collimated +sprays of hadrons that manifest the underlying struc- +ture of of quark and gluon scattering [1, 2]. From the +early days of QCD, global energy flow distributions have +been characterized using infrared and collinear safe shape +functions [3–11]. Alternatively, it can also be studied us- +ing statistical correlation of final-state energy flows, of +which the simplest one is the two-point Energy-Energy +Correlator (EEC) [12–15]. +In fact, the two different +approaches for analyzing energy flow distributions are +closely related by an integral transformation [16–19]. +In the e+e− scattering, EEC can be written as the +Fourier transformation of the correlation function of en- +ergy flow operators +EEC(y) = 8π2 +q2σ0 +� +d4x eiq·x13⟨Jµ(x1)E(n2)E(n4)J† +µ(x3)⟩ , +(1) +where n2 = (1,⃗n2) and n4 = (1,⃗n4) specify the direc- +tions of the detected energy flow in the collider, y = +1 − +(n2·n4)q2 +2(n2·q)(n4·q) = (1 + cos θ)/2, q2 > 0 is a timelike +momentum, Jµ is the electromagnetic current, and the +energy flow operator is defined as a detector time integral +of the energy-momentum tensor [20–27] +E(ni) = +∞ +� +−∞ +d ni·xi +16 +lim +¯ni·xi→∞(¯ni · xi)2Tµν(xi)¯nµ +i ¯nν +i . +(2) +Experimental studies of EEC have a long history [28–37]. +They have been used to provide precision extraction of +strong coupling constant [38, 39]. +Intuitively, when two narrow jets are produced in +e+e−, they tend to be back-to-back due to momentum +conservation, reflecting the underlying q¯q production. +This leads to a peak for EEC as y → 0, known as the +Sudakov peak. In perturbation theory, it is manifested +as large logarithmic corrections, +EEC(y) ∼ +∞ +� +n=1 +2n−1 +� +m=0 +αn +s +� +cn,m +logm y +y ++ dn,m logm y +� +, +(3) +where we have only shown the Leading Power (LP, ∼ +y−1) and the Next-to-Leading Power (NLP, ∼ y0) cor- +rections for simplicity. The leading power series of EEC +resembles the perturbative structure of vector boson pro- +duction at small pT and exhibits lightcone divergences. +It can be resummed to all orders in the perturbative cou- +pling by solving a 2d renormalization group equation in +virtuality and rapidity [40, 41]. Using the recently avail- +able 4-loop rapidity anomalous dimension [42, 43], its +perturbative resummation has been performed to N4LL +accuracy [43]. However, power corrections to EEC, and +in general to event shape functions and transverse mo- +mentum dependent observables, are much less under- +stood, both perturbatively and non-perturbatively. Re- +cently there have been significant developments towards +a more satisfying picture of power corrections for various +observables, see e.g. [44–88]. Yet the status is still far +from that of leading power terms. +In this work we initiate a study of power corrections to +EEC by exploiting conformal symmetry and techniques +from the analytic conformal bootstrap [89–92].1 Building +upon an important observation by Korchemsky [99], we +connect the logarithmically enhanced terms in power cor- +rections (m > 0 in (3)) with the expansion of correlators +around the double lightcone limit, which is controlled by +the twist expansion and the large spin expansion. Using +1 Applications of conformal symmetry in QCD has a long his- +tory [93]. For recent applications, see e.g. [42, 43, 94–98]. +arXiv:2301.03616v1 [hep-ph] 9 Jan 2023 + +2 +t +lightlike infinity +x1 +x3 +x2 +x4 +r +FIG. 1. Penrose diagram for the double lightcone limit of the +local correlator. +twist conformal blocks [100], tails from large spins can be +systematically resummed. Further simplifications come +from crossing symmetry, which relates the twist correc- +tions with large spin corrections. Using this method, we +explicitly carry out a calculation in N = 4 super Yang- +Mills (SYM) theory, and achieve the first Leading and +Next-to-Leading Logarithmic resummation at the sub- +leading power. +BACK-TO-BACK V.S. DOUBLE LIGHTCONE +As is clear from (1), EEC is related to the local Wight- +man correlator ⟨Ω|J(x1)T(x2)T(x4)J(x3)|Ω⟩ by detector +time integrals and a Fourier transform [25, 26]. It is use- +ful to understand which region of the local correlator cor- +responds to the y → 0 limit of EEC. Let us backtrack the +dominant contribution by first undoing the Fourier trans- +formation. +We can apply a Lorentz transformation to +make the two detectors exactly back-to-back: nµ +4 → ¯nµ +2 = +(1, −⃗n2). At the same time, the momentum qµ gains a +small transverse component ⃗q⊥, which is schematically +related to y as |⃗q⊥|2/q2 ∼ y. In position space, this cor- +responds to the region where |(n2·x13)(¯n2·x13)/x2 +13| ∼ y. +Since E(n2)E(¯n2) is invariant under the boost along ⃗n2, +we have an extra degree of freedom to choose a frame +such that n2 · x13, ¯n2 · x13 ∼ √y|x13|. In such a frame, +the lightcone singularities of 1 and 3 are very close on +the path of detector time integrals of 2 and 4. Therefore, +we expect the dominant contribution to come from the +region where 2 and 4 are near the pinch of lightcone sin- +gularities: x2 +12, x2 +23, x2 +34, x2 +14 ≪ x2 +13, x2 +24, which is the null +square configuration [101]. Instead, we can also choose +the frame that n2·x13 ∼ y|x13|, ¯n2·x13 ∼ |x13|. Then the +leading y → 0 dependence comes from 2 being near the +lightcone limit of both 1 and 3, which is called the double +lightcone limit. In this work we delineate this correspon- +dence and systematically go beyond the leading power +results in [99]. +It is well known that the lightcone limit is controlled by +twist expansion [102–105]. The most singular behavior +in the 1, 2-channel as x2 +12 → 0 is produced by operators +with the lowest twist τ = ∆ − ℓ. However, any single +operator in the 1, 2-channel cannot generate the lightcone +singularity in the crossed channel with x2 +13 → 0. +The +x2 +13 → 0 singularity can only be produced by an infinite +sum of operators with different spins at a given twist. +Therefore, the double lightcone limit, or relatedly the +back-to-back limit of EEC, is determined by the large +spin asymptotics of low-twist operators. +To be specific, we now consider EEC in N = 4 SYM. +Comments for QCD will be given at the end. We consider +the correlator of scalar operators belonging to the stress +tensor supermultiplet +⟨O(x1)O(x2)O(x3)O(x4)⟩dyn = +1 +(2π)4 +x4 +13x4 +24 +(x6 +12x2 +34)6 F(u, v) . +(4) +Here we have only kept the dynamical part and sub- +tracted the contribution of protected operators which are +not perturbatively corrected, see Supplemental Material +for details. +Conformal symmetry ensures that F is a +function of the conformal cross ratios +u = x2 +12x2 +34 +x2 +13x2 +24 += z¯z , +v = x2 +23x2 +14 +x2 +13x2 +24 += (1 − z)(1 − ¯z) . (5) +Expanded in the small coupling a = g2Nc +4π2 , F(u, v) reads +F(u, v) = +∞ +� +n=0 +anF(n)(u, v) = F(0)(u, v) + u3 +v Φ(u, v) , +(6) +The new function Φ(u, v) = � +n≥1 anΦ(n)(u, v) packages +all the coupling dependent information and is crossing +symmetric, i.e., Φ(u, v) = Φ(v, u). It has been calculated +to three loops in [106, 107]. The correlator admits the +following superconformal block decomposition [108, 109]: +F(u, v) = +� +∆ +� +even ℓ +aτ,ℓG∆+4,ℓ(u, v) . +(7) +Here the sums are over all superconformal primary oper- +ators with dimension ∆, spin ℓ and OPE coefficient aτ,ℓ. +The use of τ in the label foreshadows the twist expansion +later. The 4d bosonic conformal blocks are given by [110] +G∆,ℓ(u, v) = +z¯z +¯z − z [k∆−ℓ−2(z)k∆+ℓ(¯z) − (z ↔ ¯z)] , (8) +where kβ(x) = xβ/22F1( β +2 , β +2 , β; x). They are eigenfunc- +tions of the quadratic Casimir operator +D2 = z2((1 − z)∂2 +z − ∂z) + (d − 2)z¯z +z − ¯z +(1 − z)∂z + (z ↔ ¯z) +(9) + +3 +with eigenvalues +1 +2 (∆(∆ − 4) + ℓ(ℓ + 2)) [111]. +Super- +conformal symmetry causes a shift ∆ → ∆ + 4 in the +expansion (7), and we denote Gτ,ℓ(z, ¯z) = G∆+4,ℓ(u, v) +for convenience. Perturbative corrections, via conformal +block decomposition, are encoded in the expansions of +twists and OPE coefficients +τ = τ0 + +∞ +� +n=1 +anγ(n) +τ0,ℓ , +aτ,ℓ = +∞ +� +n=0 +ana(n) +τ0,ℓ , +(10) +where τ0 is the classical twist and � +n≥1 anγ(n) +τ0,ℓ = γτ,ℓ +is the anomalous dimension. The anomalous dimensions +enter via the expansion for conformal blocks: Gτ,ℓ = +�∞ +n=0 +1 +n!γn +τ,ℓ∂n +τ0Gτ0,ℓ. +Note that ∂n +τ0Gτ0,ℓ contains at +most logn u in the small u limit. +The double lightcone limit corresponds to u, v → 0, +or equivalently, z → 0, ¯z → 1. +Each conformal block +in this limit is controlled by the twist G∆,ℓ(z, ¯z) ∼ +zτ/2 log(1 − ¯z). +The presence of logarithms partially +demonstrates the effect of summing over infinitely many +spinning operators because each conformal block contains +infinitely many descendants in a given conformal family. +However, we will encounter more divergent pieces than a +single logarithm, which require the large spin contribu- +tion from infinitely many primary operators. Examples +include power divergences (1 − ¯z)m<0 and powers of log- +arithms [log(1 − ¯z)]k≥2 2. The latter arises from logk u +of ∂k +τ0Gτ0,ℓ in small u under crossing. In the following, +we refer to these as enhanced divergences. To lighten the +notations, we will denote the logarithms as log(x) = Lx +from now on. +We also point out that crossing symmetry will play an +important role in our computation of the power correc- +tions. As we will see, the u, v power corrections to Φ(u, v) +contribute equally to the y power corrections in EEC. +In particular, EEC at NLP corresponds to order u0v1 +and u1v0 in Φ(u, v). +The former only requires twist-2 +data to subleading order in the large spin limit (O(ℓ−2)), +while the latter contains twist-4 contributions. Crossing +symmetry equates these two contributions and therefore +avoids the necessity of inputting the twist-4 information. +TABLE I. CFT data needed at different orders +Power Corrections Perturbative Corrections +twist +large spin +LL +NLL +LP +2 +O(ℓ0) +a(0) +2,ℓ , γ(1) +2,ℓ +a(1) +2,ℓ , γ(2) +2,ℓ +NLP +2 +O(ℓ−2) +a(0) +2,ℓ , γ(1) +2,ℓ +a(1) +2,ℓ , γ(2) +2,ℓ +4 +O(ℓ0) +a(0) +4,ℓ , γ(1) +4,ℓ +a(1) +4,ℓ , γ(2) +4,ℓ +2 At one loop the perturbative logarithms at NLP have at most +k = 1, therefore are not fixed by large spins. +Before going into the technical details, we provide a +brief overview for the next two sections. We will first use +techniques from large spin perturbation theory to extract +the enhanced divergences in v at twist-2 up to NLP. That +is, we will obtain Φ(u, v) = p0(Lu, Lv)+p1(Lu, Lv)v+· · · , +in which p0, p1 are polynomials in Lu, Lv at each pertur- +bative order. Crossing symmetry then fixes the NLP con- +tribution in u to be Φ(u, v) = p0(Lu, Lv)+p1(Lu, Lv)v + +p1(Lv, Lu)u · · · . The relevant data is listed in Table I, +but only the first two rows are needed thanks to crossing +symmetry. Finally, we map the small u, v expansion of +Φ(u, v) to the back-to-back limit of EEC(y). The rules at +LP and NLP are given in Table II and are valid to NLL +accuracy. We will explain these points in more detail in +the rest of the paper. +TABLE II. Logarithms Map at LP and NLP +Φ(u, v) 4y(1 − y)2 × EEC(y) +LP +Lm +u Lm +v +2(m + n)Lm+n−1 +y/(1−y) +NLP uLm +u Ln +v +2m(1−m) +m+n−1 +y +1−y Lm+n−1 +y/(1−y) +vLm +u Ln +v +2n(1−n) +m+n−1 +y +1−y Lm+n−1 +y/(1−y) +LARGE SPIN ANALYSIS +The enhanced divergences can be systematically han- +dled by using the Large Spin Perturbation Theory [100], +which culminates an array of earlier works in the large +spin sector [112–118]. The starting point is the free the- +ory limit where the twists are degenerate. The correlators +can be written as a sum over the twists +F(0)(z, ¯z) = +� +τ0=2,4,... +Hτ0(z, ¯z) , +(11) +where each Hτ0(z, ¯z) sums over spins +Hτ0(z, ¯z) = +∞ +� +ℓ=0 +⟨a(0) +τ0,ℓ⟩Gτ0,ℓ(z, ¯z) , +(12) +and is known as a twist conformal block (TCB). When +the interaction is turned on, the twist degeneracies are +lifted and the OPE coefficients are corrected. In general, +we find that the expansion consists of sums of the form +∞ +� +ℓ=0 +⟨a(0) +τ0,ℓ⟩κτ0(ℓ)Gτ0,ℓ(z, ¯z) , +(13) +where the quantities κτ0(ℓ) admit expansions around +large conformal spins J2 +τ,ℓ = (ℓ + τ +2)(ℓ + τ +2 − 1) with the +following schematic form +κτ0(ℓ) = +∞ +� +m=0 +N +� +i=0 +Km,i +J2m +˜τ0,ℓ +logi J2 +˜τ0,ℓ . +(14) + +4 +Here we introduce shifted twists ˜τ0 = τ0 + 4 to take into +account the dimension shift in the decomposition (7). An +interesting feature, as we will see in explicit calculations, +is that only even negative powers appear in the expan- +sion. This is closely related to the reciprocity relation +[119, 120], which has been explicitly verified in QCD to +three loops [121]. +Consequently, the correlator should +now be expanded in terms of a more general class of +TCBs +H(m,i) +τ0 +(z, ¯z) = +� +ℓ +a(0) +τ0,ℓ +logi J2 +˜τ0,ℓ +J2m +˜τ0,ℓ +Gτ0,ℓ(z, ¯z) . +(15) +Note that conformal blocks satisfy the Casimir equation +C˜τ0Gτ0,ℓ(z, ¯z) = J2 +˜τ0,ℓGτ0,ℓ(z, ¯z) , +(16) +where Cτ = D2 + 1 +4τ(2d − τ − 2) is the shifted confor- +mal Casimir. It follows that TCBs obey the following +recursion relations +H(m,i) +τ0 +(z, ¯z) = C˜τ0H(m+1,i) +τ0 +(z, ¯z) . +(17) +The full TCBs are in general difficult to compute. How- +ever, we will only need them in the small v limit where +computations become manageable. The TCBs in 4D take +a factorized form +H(m,i) +τ0 +(z, ¯z) = z k˜τ0−2(z) +¯z − z +¯H(m,i) +τ0 +(¯z) , +(18) +where we have dropped the regular part when ¯z → 1. +Moreover, the recursion relation (17) becomes +¯H(m,i) +τ0 +(¯z) = ¯D ¯H(m+1,i) +τ0 +(¯z) , +(19) +where +¯D = ¯z2(1 − ¯z) d2 +d¯z2 − ¯z(2 − ¯z) d +d¯z + 2 − ¯z . +(20) +Using this recursion relation, one can compute the TCBs +explicitly in the small v limit [122]. For example, we find +¯H(0,i) +2 +(¯z) = (−1)i +�Li +ϵ +2ϵ + +Li+1 +ϵ +6(i + 1) + γE − 3 +3 +Li +ϵ +� ++ · · · , +¯H(1,i) +2 +(¯z) = (−1)i +� +Li+2 +ϵ +2(i + 1)(i + 2) + γELi+1 +ϵ +i + 1 +� ++ · · · .(21) +where we have defined ϵ = 1 − ¯z. +Let us now focus on the small u limit, i.e., z → 0. +Together with ¯z → 1, the correlator takes the form +F(n) = z3 logn z +ϵ +� +logn ϵ(An,1 + An,2ϵ + . . .) ++ logn−1 ϵ(Bn,1 + Bn,2ϵ + . . .) + . . . +� ++ O(z4) += z3 +∞ +� +m=0 +n +� +i=0 +Cm,i ¯H(m,i) +2 +(¯z) + O(z4) . +An important point is that to compute the NqLP in the +small ϵ expansion, i.e., An,j=1,2,...,q, Bn,j=1,2,...,q etc, we +only need ¯H(m,i) +2 +with m = 0, 1, . . . , q. To see this, we +act on the two expansions with ¯D and compare the power +divergences. Note that for any polynomial p(ϵ) +¯D(p(ϵ) logi ϵ) = i(i − 1)(1 − ϵ)p(ϵ) logi−2 ϵ +ϵ ++ O(ϵ0) . +Taking p(ϵ) = ϵq at the q-th order, we find the RHS +only becomes a power divergence after acting q times +with ¯D. On the other hand, it is known that only the +TCBs with m ≤ 0 are power divergent. The repeated ¯D +action makes the TCBs with m ≤ q power divergent and +therefore responsible for the q-th order correction. +We now explicitly compute the power corrections using +the TCB decomposition. From Table I, to NLL and 2nd +order in NLP the needed data is [122–124] +a(0) +2,ℓ = Γ(ℓ + 3)2 +Γ(2ℓ + 5) , +(22) +γ(1) +2,ℓ = log J2 +6,ℓ + 2γE + +1 +3J2 +6,ℓ ++ O(J−4 +6,ℓ ) , +(23) +a(1) +2,ℓ +a(0) +2,ℓ += +�1 + 4J6,ℓ +16J2 +6,ℓ +−log 2 +� +γ(1) +2,ℓ − ζ2+ 1 +J6,ℓ ++O(J−3 +6,ℓ ),(24) +γ(2) +2,ℓ = +� 1 +J6,ℓ +− ζ2 +2 +� +γ(1) +2,ℓ − 3ζ3 +2 + +1 +J2 +6,ℓ ++ O(J−3 +6,ℓ ) .(25) +From the conformal block decomposition (7), the small +u expansion of F(n) up to NLL accuracy reads +F(n) = z3 � +even ℓ +a(0) +2,ℓ +�� +γ(1) +2,ℓ +�n +2nn! +Ln +z + +� +γ(1) +2,ℓ +�n−1 +Ln−1 +z +2n−1(n − 1)! +× +� +a(1) +2,ℓ +a(0) +2,ℓ ++ (n − 1) +γ(2) +2,ℓ +γ(1) +2,ℓ ++ +γ(1) +2,ℓ ∂ℓ +2 +� � +k2ℓ+6(¯z) + · · · , +(26) +where all the odd powers in 1/J6,ℓ cancel out upon using +the large spin expansion of CFT data (23-25). We can +therefore rewrite it in terms of TCBs as +F(n) = +(27) +Ln +z +n! +� +i +�n +i +� �γn−i +E +2i H(0,i) +2 ++ n − i +3 +γn−1−i +E +2i+1 H(1,i) +2 +� ++ · · · . +We have only showed the LL part for brevity and left the +NLL part to Supplemental Material. Substituting in the +TCBs using (18) and (21) we obtain F(n) at LP in z and +NLP in 1 − ¯z +F(n) = (−1)nz3 +2n+1n! Ln +z Ln +ϵ +�1 − ϵ +ϵ ++ 2n +3Lϵ ++ 1 +Lz ++· · · +� ++O(z4) , +(28) + +5 +with NLL accuracy. Using F(n)(z, ¯z) = +v +u3 Φ(n)(u, v) and +crossing symmetry Φ(u, v) = Φ(v, u), we get the NLL +prediction for Φ(n) at NLP in the double lightcone limit +Φ(n) = (−1)n +2nn! Ln +uLn +v +�1 +2 + (u + v) +(29) ++ +��n + 1 +2 +u + n +3 v +� 1 +Lv ++ (u ↔ v) +� ++ · · · +� +, +n > 1 . +As was promised in the last section, only twist-2 CFT +data was used in the whole process. This prediction is +checked against the available two- and three-loop results +in the Supplemental Material. +POWER CORRECTIONS TO EEC IN N = 4 SYM +The final task is to find the explicit relation between +the double lightcone limit series uj1vj2Lm +u Ln +v and the +back-to-back limit series yjLk +y. The answer can be found +using the Mellin representation of EEC [25, 26, 125] +EEC(y) = +1 +4y(1 − y)2 +� +dj1dj2 +(2πi)2 M(j1, j2)K(j1, j2; y) , +(30) +where M(j1, j2) is the Mellin amplitude for Φ(u, v): +Φ(u, v) += +� dj1dj2 +(2πi)2 M(j1, j2)uj1vj2 +and +the +kernel +K(j1, j2; y) is defined as +K(j1, j2; y) = +2Γ(1 − j1 − j2) +� +y +1−y +�j1+j2 +Γ(j1 + j2) [Γ(1 − j1)Γ(1 − j2)]2 . +(31) +This gives the map uj1vj2 → K(j1,j2;y) +4y(1−y)2 . Taking deriva- +tives w.r.t. +j1, j2 generates the rules containing loga- +rithms in u, v. One subtlety is that, due to the presence +of the pole at j1 + j2 = 1 in K(j1, j2; y), the maps at +NLP cannot predict the y0 term without any log y en- +hancement. But it can be shown that all the logarithmic +contributions at NLP are preserved (see Supplemental +Material). +The rules at LP and NLP up to NLL are +summarized in Table II, and lead to +EEC(n>1)(y) = +(−1)n +2n(n − 1)! +� 1 +2y +� +L2n−1 +y ++ O(L2n−3 +y +) +� ++ +� +n +2n − 1L2n−1 +y ++ 7n − 5 +12 +L2n−2 +y ++ O(L2n−3 +y +) +� ++ · · · +� +, +(32) +which is in full agreement with the full theory calculation +up to n = 3 in [126]. The n > 3 terms are new and are +one of the main result of this work. The analytic series +in n can be resummed explicitly to all orders, leading to +160 +165 +170 +175 +180 +0.0 +0.2 +0.4 +0.6 +0.8 +θ +EEC(θ) +EEC Back-to-Back Limit Resummation +FIG. 2. EEC as a function of θ in the back-to-back limit. We +use g2/(4π) = 0.118 to mimic the QCD strong coupling at Z +pole. The dashed line refers to LP resummed to NLL, with +the inclusion of NLP terms up to NNLO (n ≤ 3). +the following NLL formula at LP and NLP3 +EEC(y) = −aLye− +aL2 +y +2 +4y +− 1 +4 +��π +2 +√a erf +��a +2Ly +� ++aLye− +aL2 +y +2 +� ++ a +48(7aL2 +y − 4)e− +aL2 +y +2 ++ a +12 + · · · , (33) +where erf is the error function erf(x) = +2 +√π +� x +0 e−t2dt 4. +In Fig. 2 we plot the N = 4 EEC in the back-to-back +limit to illustrate the importance of NLP resummation. +It can be seen that the LL and NLL series at NLP leads to +substantial corrections for not too large θ. For θ > 175◦ +the Sudakov double logs suppressed the NLP contribu- +tions. +For comparison we also plot in dashed line the +fixed-order NLP results truncated to NNLO [126], along +with the LP NLL series. In this case sizable NLP cor- +rections can be found for θ > 175◦, which however is +misleading as they disappear after resumming to all or- +ders in coupling. +DISCUSSIONS +Our results lead to several exciting research avenues. +First of all, it is interesting to apply the results to EEC +in QCD, where fixed-order data up to NLO has become +3 The one-loop NLL contribution at NLP has no Ly enhancement. +Therefore, we need to input the one-loop EEC to fix the constant +a/12. +4 We note that the LL-NLP series has been studied in [69]. Their +results disagree with ours starting from O(a3), and seems to be +in conflict with the fixed-order analytic result in [126]. Further +comparison is provided in the Supplemental Material. + +6 +available recently [127–129]. The local correlator of four +electromagnetic currents in QCD has also been computed +at one loop [130]. Secondly, in QCD running coupling +corrections will modify NLL series. It would be impor- +tant to understand how to incorporate these effects while +retaining the power of conformal symmetry. +Thirdly, +our results provide concrete data for quantitative com- +parison between additive and multiplicative scheme in +resummation matching, see e.g. +[131]. +Fourthly, local +correlators exhibit other interesting limits, such as the +Regge limit [132]. It would be interesting to understand +what constraints are imposed on EEC by such limits. +Last but not least, it would be worthwhile to under- +stand the relation between our approach and the conven- +tional approach based on momentum space renormaliza- +tion group, in particular the relation between crossing +symmetry for local correlator and the consistency rela- +tions from infrared poles cancellations [48]. +We thank Zhongjie Huang, Kai Yan, and Xiaoyuan +Zhang for useful discussions. 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It is convenient to keep track of the R-symmetry information by contracting +the indices with null polarization vectors YI or with a traceless symmetric tensor SIJ 5 +O(x, Y ) ≡ OIJ(x)YIYJ , +or +O(x, S) = OIJ(x)SIJ . +(S.2) +Due to superconformal symmetry, the four-point function has following “partially non-renormalized” form [133] +⟨O(x1, Y1)O(x2, Y2)O(x3, Y3)O(x4, Y4)⟩ = G(0)(1, 2, 3, 4) + 2(N 2 +c − 1) +(4π2)4 +y4 +12y4 +34 +x2 +12x2 +34x2 +14x2 +23 +R Φ(u, v) +(S.3) +where G(0)(1, 2, 3, 4) is the tree-level correlator +G(0)(1, 2, 3, 4) =(N 2 +c − 1)2 +4(4π2)4 +�� y2 +12y2 +34 +x2 +12x2 +34 +�2 ++ +� y2 +13y2 +24 +x2 +13x2 +24 +�2 ++ +� y2 +41y2 +23 +x2 +41x2 +23 +�2� ++N 2 +c − 1 +(4π2)4 +� y2 +12y2 +23y2 +34y2 +41 +x2 +12x2 +34x2 +23x2 +41 ++ y2 +12y2 +24y2 +34y2 +13 +x2 +12x2 +24x2 +34x2 +31 ++ y2 +13y2 +23y2 +24y2 +41 +x2 +13x2 +23x2 +24x2 +41 +� +, +(S.4) +with y2 +ij ≡ Yi · Yj and the function Φ(u, v) encodes all the dynamical information. The factor R is determined by +superconformal symmetry +R = (1 − zα)(1 − z¯α)(1 − ¯zα)(1 − ¯z¯α) , +(S.5) +where the conformal cross ratios z, ¯z have already been introduced in the main text and α, ¯α are similarly the +R-symmetry cross ratios defined by +y2 +13y2 +24 +y2 +12y2 +34 += α¯α , +y2 +14y2 +23 +y2 +12y2 +34 += (1 − α)(1 − ¯α) . +(S.6) +The weak coupling g ≪ 1 expansion of Φ(u, v) reads +Φ(u, v) = +∞ +� +n=1 +anΦ(n)(u, v) , +a = g2Nc +4π2 , +(S.7) +and is known up to three loops [107]. Since the dynamic function Φ(u, v) is R-symmetry independent, we can choose +special polarizations to simplify the correlator (S.3). Following [25], let us take +YI = (1, 0, 1, 0, i, i) , +SIJ = diag(1, −1, 0, 0, 0, 0) , +S′ +IJ = diag(0, 0, 1, −1, 0, 0) , +(S.8) +and define +� +O(x2) = 2O(x2, S) , +� +O′(x4) = 2O(x4, S′) , +O(x3) = +�N 2 +c − 1 +2π4 +�− 1 +2 +O(x3, Y ) , +O†(x1) = +�N 2 +c − 1 +2π4 +�− 1 +2 +O(x1, Y ∗) . +(S.9) +5 The tensor can be built from the vectors, e.g., SIJ = Y ′ +I Y ′ +J + +Y ′′ +I Y ′′ +J . Therefore, it is sufficient to focus on the former case. + +10 +Then the four-point function becomes +⟨O†(x1) � +O(x2) � +O′(x4)O(x3)⟩ = +1 +(2π)4 +1 +(x2 +12x2 +34)2 +�N 2 +c − 1 +8 +� +1 + u2 +v2 +� ++ u +v +�1 +2 + Φ(u, v) +�� += +1 +(2π)4 +1 +(x2 +12x2 +34)2 +�N 2 +c − 1 +8 +Gshort(u, v) + 1 +u2 F(u, v) +� +, +(S.10) +where we have further split it into short multiplet contribution Gshort(u, v) and the long multiplet contribution F(u, v). +The explicit form of Gshort(u, v) can be found in [109] and is protected from perturbative corrections. As a result, +Φ(u, v) is essentially the same as all the loop corrections (n ≥ 1) of F(u, v) = � +n≥0 anF(n)(u, v), i.e., +F(u, v) − F(0)(u, v) = u3 +v Φ(u, v) . +(S.11) +Another reason for choosing the polarizations (S.8) is the relation to the scalar detectors +S(ni) = 1 +4 +∞ +� +−∞ +dni · xi +lim +¯ni·xi→∞(¯ni · xi)2O(xi) . +(S.12) +Thanks to superconformal symmetry, the spinning correlator ⟨JTTJ⟩ is related to the scalar correlator (S.3) by Ward +identities. Moreover, the EEC and the scalar-scalar correlation (SSC) are also proportional [25] +⟨E(n2)E(n4)⟩ = +4(q2)2 +(n2 · n4)2 ⟨S(n2)S(n4)⟩ . +(S.13) +Twist Conformal Blocks +The TCBs with logarithms can be computed using the method of [122]. The idea is to apply the recursion relations +on ¯H(0,0) +τ0 +(¯z), which is determined by the tree-level correlator, and compute ¯H(m,0) +τ0 +(¯z) at negative integer m. We then +analytic continue in m and take derivatives to obtain the logarithms. For our case, τ0 = 2 and we have +¯H(0,0) +2 +(¯z) = ¯z2(2 − ¯z) +2(1 − ¯z) + ¯z log(1 − ¯z) = 1 +2ϵ + regular terms . +(S.14) +Repeated ¯D action gives ¯H(m,0) +2 +(¯z) and the first few terms in small ϵ for negative integer m are given by (3.37) of +[122] +¯H(m,0) +2 +(¯z) =1 +2ϵm−1Γ(1 − m)2 + 1 +6m +� +2m2 − 6m + 1 +� +ϵmΓ(−m)2 ++ +1 +180(m − 1)m(m + 1) +� +20m3 − 54m2 − 35m + 36 +� +ϵm+1Γ(−m − 1)2 + · · · . +(S.15) +Obtaining the full analytic expression for ¯H(m,i) +2 +(ϵ) is difficult. But life is much easier if we content ourselves with +getting first few orders in ϵ and logarithms. Truncated to order ϵ0, only ¯H(0,i) +2 +(ϵ) and ¯H(1,i) +2 +(ϵ) are relevant and we +find +¯H(0,i) +2 +(¯z) = (−1)i +ϵ +�1 +2 logi ϵ + iγELi−1 +ϵ ++ i(i − 1) +12 +(12γ2 +E + π2)Li−2 +ϵ ++ · · · +� +(S.16) ++ (−1)i +� +1 +6(i + 1)Li+1 +ϵ ++ γE − 3 +3 +Li +ϵ + π2 + 12γ2 +E − 72γE + 12 +36 +iLi−1 +ϵ ++ . . . +� ++ · · · , +¯H(1,i) +2 +(¯z) = (−1)i +� +1 +2(i + 1)(i + 2)Li+2 +ϵ ++ γE +i + 1Li+1 +ϵ ++ 12γ2 +E + π2 +12 +Li +ϵ + · · · +� ++ · · · . +(S.17) + +11 +More Details of (26) and (27) +In this section, we present the details of the large spin perturbation calculation needed for obtaining EEC in the +back-to-back limit to the NLL and NLP order. As we explained in the main text, only twist-2 contributions are +needed. The expansion of the twist-2 conformal block G2+γ2,ℓ,ℓ(z, ¯z) in the z → 0 limit is +G2+γ2,ℓ,ℓ(z, ¯z) = z3+γ2,ℓ/2k6+2ℓ+γ2,ℓ(¯z) + O(z4) += z3 +∞ +� +n=0 +an +� � +1 +2nn! +� +γ(1) +2,ℓ +�n +logn z + +1 +2n−1(n − 2)!γ(2) +2,ℓ +� +γ(1) +2,ℓ +�n−2 +logn−1 z + · · · +� +k6+2ℓ(¯z) ++ +1 +2n(n − 1)! +� +γ(1) +2,ℓ +�n +logn−1 z ∂ℓk6+2ℓ(¯z) + · · · +� ++ O(z4) . +(S.18) +Combined with the expansion of the OPE coefficient a2,ℓ, we obtain the leading twist contribution to F(z, ¯z) +F(n)(z, ¯z) = z3 � +even ℓ +� +logn z +1 +2nn!a(0) +2,ℓ +� +γ(1) +2,ℓ +�n +k6+2ℓ(¯z) + logn−1 z +� +a(1) +2,ℓ +� +γ(1) +2,ℓ +�n−1 +2n−1(n − 1)! + a(0) +2,ℓ +γ(2) +2,ℓ +� +γ(1) +2,ℓ +�n−2 +2n−1(n − 2)! +� +k6+2ℓ(¯z) ++ logn−1 z a(0) +2,ℓ +1 +2n(n − 1)! +� +γ(1) +2,ℓ +�n +∂ℓk6+2ℓ(¯z) + · · · +� ++ O(z4) , +(S.19) +which is NLL in log z. Then using the IBP identity +a(0) +2,ℓ∂ℓk6+2ℓ(¯z) = ∂ℓ[a(0) +2,ℓk6+2ℓ(¯z)] − k6+2ℓ(¯z)∂ℓa(0) +2,ℓ , +(S.20) +and a(1) +2,ℓ = −ζ2a(0) +2,ℓ + 1 +2∂ℓ +� +a(0) +2,ℓγ(1) +2,ℓ +� +, we rewrite (S.19) as +F(n)(z, ¯z) =z3 � +even ℓ +� +1 +2nn! logn z a(0) +2,ℓ +� +γ(1) +2,ℓ +�n +k6+2ℓ(¯z) + +1 +2n(n − 1)! logn−1 z +� � +γ(1) +2,ℓ +�n +∂ℓ +� +a(0) +2,ℓk6+2ℓ(¯z) +� ++a(0) +2,ℓk6+2ℓ(¯z) +� +γ(1) +2,ℓ +�n−2 �� +−2ζ2 + ∂ℓγ(1) +2,ℓ +� +γ(1) +2,ℓ + 2(n − 1)γ(2) +2,ℓ +� � ++ · · · +� ++ O(z4) . +(S.21) +The use of (S.20) becomes clear when we use the integer-step finite difference to approximate ∂ℓ +� +a(0) +2,ℓk6+2ℓ(¯z) +� +at +large spin ℓ. On a general function f(ℓ), we approximate f ′(ℓ) ≈ f(ℓ+2)−f(ℓ−2) +4 +, which is accurate up to O(ℓ−2). +Neglecting boundary terms which vanish at large spins, we can write +� +even ℓ +� +γ(1) +2,ℓ +�n +∂ℓ +� +a(0) +2,ℓk6+2ℓ(¯z) +� +≈ 1 +4 +� +even ℓ +a(0) +2,ℓk6+2ℓ(¯z) +�� +γ(1) +2,ℓ−2 +�n +− +� +γ(1) +2,ℓ+2 +�n� +, +(S.22) +Expanding everything other than a(0) +2,ℓ with respect to the large conformal spin J2 +6,ℓ, we get +F(n)(z, ¯z) = z3 � +even ℓ +� +1 +2nn! logn z a(0) +2,ℓk6+2ℓ(¯z) +� +� +log(J2 +6,ℓ) + 2γE +�n + n +3 +� +log(J2 +6,ℓ) + 2γE +�n−1 +1 +J2 +6,ℓ ++ · · · +� ++ +1 +2n(n − 1)! logn−1 z a(0) +2,ℓk6+2ℓ(¯z) +� +− (n + 1)ζ2 +� +log(J2 +6,ℓ) + 2γE +�n−1 +− 3(n − 1)ζ3 +� +log(J2 +6,ℓ) + 2γE +�n−2 + +1 +J2 +6,ℓ +� +(n − 1) +� +1 − ζ2 +3 (n + 1) +� � +log(J2 +6,ℓ) + 2γE +�n−2 +−(n − 1)(n − 2)ζ3 +� +log(J2 +6,ℓ) + 2γE +�n−3 �� ++ · · · +� ++ O(z4) , +(S.23) + +12 +which can be organized into TCBs as +F(n)(z, ¯z) =logn z +2nn! +� n +� +i=0 +n! +i!(n − i)!(2γE)iH(0,n−i) +2 ++ n +3 +n−1−i +� +i=0 +(n − 1)! +i!(n − 1 − i)!(2γE)iH(1,n−1−i) +2 ++ · · · +� ++ logn−1 z +2n(n − 1)! +� +− (n + 1)ζ2 +n−1 +� +i=0 +(n − 1)! +i!(n − 1 − i)!(2γE)iH(0,n−1−i) +2 +− 3(n − 1)ζ3 +n−2 +� +i=0 +(n − 2)! +i!(n − 2 − i)!(2γE)iH(0,n−2−i) +2 ++ (n − 1) +� +1 − ζ2 +3 (n + 1) +� n−2 +� +i=0 +(n − 2)! +i!(n − 2 − i)!(2γE)iH(1,n−2−i) +2 +− (n − 1)(n − 2)ζ3 +n−3 +� +i=0 +(n − 3)! +i!(n − 3 − i)!(2γE)iH(1,n−3−i) +2 ++ · · · +� ++ · · · . +(S.24) +Substituting the explicit TCBs (S.16, S.17), we get +F(n)(z, ¯z)= z3 +� 1 +n! logn z +�1 +ϵ +�(−1)n +2n+1 logn ϵ + · · · +� ++ +�(−1)n+1 +2n+1 +logn ϵ + (−1)nn +3 × 2n logn−1 ϵ + · · · +� ++ · · · +� ++logn−1 z +(n − 1)! +�1 +ϵ +�(−1)n +2n+1 (n + 1)ζ2 logn−1 ϵ − (−1)n +2n+1 3(n − 1)ζ3 logn−2 ϵ + · · · +� ++ +�(−1)n +2n+1n logn ϵ + (−1)n+1 +2n+1 +(n + 1)ζ2 logn−1 ϵ + · · · +� � ++ · · · +� ++ O(z4) .(S.25) +Via F(n)(z, ¯z) = +v +u3 Φ(n)(z, ¯z), this gives the expansion of Φ(n)(z, ¯z). Crossing symmetry allows us to further recon- +struct the O(u1v0) contributions, which gives the results in (29). +Details of the Map from uj1vj2Lm +u Ln +v to yjLk +y +In this section, we provide more details for establishing the map from the small u, v expansion of Φ(u, v) to the +small y expansion of EEC(y). Instead of electromagnetic current sources Jµ, we consider two scalar operator sources, +belonging to the stress tensor multiplet, in the center of mass frame qµ = (Q, 0, 0, 0). EEC(y) relates to ⟨E(n2)E(n4)⟩ +by an overall factor: +EEC(y) = +� +dΩ2dΩ4δ(⃗n2 · ⃗n4 − cos θ)⟨E(n2)E(n4)⟩ +Q2 += 8π2 +Q2 ⟨E(n2)E(n4)⟩ , +(S.26) +where θ is the angle between ⃗n2 and ⃗n4 and we assume the convention that ⟨E(n2)E(n4)⟩ has already been normalized +to the cross section. +The superconformal Ward identities further reduce the EEC ⟨E(n2)E(n4)⟩ to scalar-scalar +correlation (SSC) ⟨S(n2)S(n4)⟩ [25, 125] +⟨E(n2)E(n4)⟩ = +4(q2)2 +(n2 · n4)2 ⟨S(n2)S(n4)⟩ . +(S.27) +The SSC is related to the local correlator in a simple way in Mellin space [25] +Φ(u, v) = +� +dj1dj2 +(2πi)2 M(j1, j2)uj1vj2 . +(S.28) +Here M(j1, j2) is the Mellin amplitude and encodes all the dynamical information. To compute the SSC, the first +step is to obtain the Lorentzian correlator. This is achieved by using the Wightman prescription x2 +ij → −x2 +ij + iϵtij, +if operator i sits before j. In our case, the operator ordering is 1 < 2 < 4 < 3. The second step is to perform the light +transform on the Lorentzian correlator +� +i=2,4 +� ∞ +−∞ +d(ni · xi) +lim +¯ni·xi→∞ +� ¯ni · xi +2 +�2 +, +(S.29) + +13 +which turns local operators into detectors. +To obtain SSC in the momentum space, the last step is the Fourier +transformation +� +d4x13 exp(iq · x13). After normalized to the total cross section, the Mellin representation for SSC is +⟨S(n2)S(n4)⟩ = +1 +(2π)4 +π2 +(n2 · n4)q2 +1 − y +y +� +dj1dj2 +(2πi)2 M(j1, j2)K(j1, j2; y) , +(S.30) +with +K(j1, j2; y) = +2Γ(1 − j1 − j2) +Γ(j1 + j2) [Γ(1 − j1)Γ(1 − j2)]2 +� +y +1 − y +�j1+j2 +. +(S.31) +Then using n2 · n4 = 2(1 − y) in the center of mass frame, we obtain the Mellin representation for EEC(y) +EEC(y) = 8π2Q2 +(1 − y)2 ⟨S(n2)S(n4)⟩ = +1 +4y(1 − y)2 +� +dj1dj2 +(2πi)2 M(j1, j2)K(j1, j2; y) . +(S.32) +Therefore, we expect the following mapping from the double lightcone limit to back-to-back limit +uj1vj2 → K(j1, j2; y) +4y(1 − y)2 . +(S.33) +At LP and NLP, we need the following rules containing logarithms in u, v +logm u logn v → m! n! +� +|j1|=ϵ +dj1 +2πi +1 +jm+1 +1 +� +|j2|=ϵ +dj2 +2πi +1 +jn+1 +2 +K(j1, j2; y) +4y(1 − y)2 , +(S.34) +u logm u logn v → m! n! +� +|j1−1|=ϵ +dj1 +2πi +1 +(j1 − 1)m+1 +� +|j2|=ϵ +dj2 +2πi +1 +jn+1 +2 +K(j1, j2; y) +4y(1 − y)2 , +(S.35) +v logm u logn v → m! n! +� +|j1|=ϵ +dj1 +2πi +1 +jm+1 +1 +� +|j2−1|=ϵ +dj2 +2πi +1 +(j2 − 1)n+1 +K(j1, j2; y) +4y(1 − y)2 . +(S.36) +For the cases we will inspect, the only exception is the constant term at NLP which is caused by the j1 + j2 = 1 +pole in K(j1, j2; y). To see this, we consider the first derivative of 1−y +y K(j1, j2; y) w.r.t. +y +1−y. Such an action shifts +Γ(1−j1 −j2) to Γ(2−j1 −j2), which is analytic at j1 +j2 = 1. The constant term is killed after taking the derivative, +while the logarithmic information remains. The explicit expressions for general m, n up to NLL accuracy are shown +in Table II. +Comparison with Existing Results +In this section we provide a comparison of our predictions with the existing results in the literature. We begin +with the local correlator in the double lightcone limit to NLL accuracy. The full three-loop correlator can be found +in [107]6. Upon expanding their results in the double lightcone limit we find +Φ(1)(u, v) = +� +−1 +4 log u log v + 0 · log(uv) + · · · +� +− +�1 +4(u + v) log u log v + 1 +2(u log u + v log v) + · · · +� ++ · · · , +Φ(2)(u, v) = +� 1 +16 log2 u log2 v + 0 · log u log v log(uv) + · · · +� ++ +�1 +8(u + v) log2 u log2 v ++ 3 +16 log u log v(u log u + v log v) + 1 +8 log u log v(v log u + u log v) + · · · +� ++ · · · , +Φ(3)(u, v) = +� +− 1 +96 log3 u log3 v + 0 · log2 u log2 v log(uv) + · · · +� +− +� 1 +48(u + v) log3 u log3 v ++ 1 +24 log2 u log2 v(u log u + v log v) + 1 +48 log2 u log2 v(v log u + u log v) + · · · +� ++ · · · . +(S.37) +6 There is an overall normalization difference +� +− 1 +4π +�n at the n-th +loop. + +14 +Compared with our NLL prediction (29), we find perfect agreement except for the NLP terms at one loop and a +single NLL-NLP term at two loops (shown in red explicitly). Such terms do not correspond to enhanced divergence +in v and hence cannot be captured by the large spin analysis. Moreover, the two-loop NLL-NLP mismatched term +1 +8 log u log v(v log u + u log v) does not map to NLL-NLP term in EEC, as can be checked from Table II. +For EEC in N = 4 SYM results up to three loops with full y dependence have been computed in [126], using the +local correlator from [107] as input. The back-to-back expansion of [126] up to NLL at NLP is +EEC(1) = − 1 +4y log y−1 +2 log y+0 · y0 + O(y) , +EEC(2) = 1 +y +�log3 y +8 ++ 0 · log2 y + · · · +� ++ +�log3 y +6 ++ 3 +16 log2 y + · · · +� ++ O(y) +EEC(3) = 1 +y +� +−log5 y +32 ++ 0 · log4 y + · · · +� ++ +� +−3 log5 y +80 +− log4 y +12 ++ · · · +� ++ O(y) . +(S.38) +To obtain (S.38), it is necessary to expand the following integral to NLP, +E = +� 1 +0 +d¯z +� ¯z +0 +dt +−1 +t(1 − y − ¯z) + y¯z +� +z¯z +1 − z − ¯z P1 + +z2¯z +(1 − z)2(1 − z¯z)P2 +� +, +(S.39) +where z = (1−y)t(t−¯z)/(t(1−y−¯z)+y¯z) and P1 and P2 are lengthy combinations of weight-3 harmonic polylogarithms +and can be found in [126]. The expansion of this integral begins from NLP and reads +E = log5 y +60 ++ 0 · log4 y + 1 +12ζ2 log3 y + · · · +(S.40) +where we have neglected terms of O(log2 y) and beyond. The expansion of this integral was also performed in [69], +but the result reported in Eq. (5.17) of [69] is larger than (S.40) by a factor of two. Using (S.40) and expanding the +remaining results in [126] with the package HPL, we obtain EEC(3) presented in (S.38). +Our results in (32) for n > 1 are in full agreement with EEC(2) and EEC(3) in (S.38). This provides a strong check +for our results. Our large spin analysis does not expect to capture the NLP terms at one loop (shown in red in (S.38)), +for the same reason as explained for local correlator. +We note that a LL study at NLP has also been performed in [69], where an RG equation at NLP has been derived. +After changing to our normalization, their predicted LL coefficient at NLP and three loop, given in Eq. (5.21) of [69], +is −1/30 log5 y, which disagrees with our results in (33), as well as with the full results from [126]. + diff --git a/UNE2T4oBgHgl3EQfCwYa/content/tmp_files/load_file.txt b/UNE2T4oBgHgl3EQfCwYa/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a7f281984063edb618d12f64e9e88107feabf2f2 --- /dev/null +++ b/UNE2T4oBgHgl3EQfCwYa/content/tmp_files/load_file.txt @@ -0,0 +1,1424 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf,len=1423 +page_content='Power Corrections to Energy Flow Correlations from Large Spin Perturbation Hao Chen,1, ∗ Xinan Zhou,2, † and Hua Xing Zhu1, ‡ 1Zhejiang Institute of Modern Physics, School of Physics, Zhejiang University, Hangzhou, 310027, China 2Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Dynamics of high energy scattering in Quantum Chromodynamics (QCD) are primarily probed through detector energy flow correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' One important example is the Energy-Energy Corre- lator (EEC), whose back-to-back limit probes correlations of QCD on the lightcone and can be described by a transverse-momentum dependent factorization formula in the leading power approxi- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' In this work, we develop a systematic method to go beyond this approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' We identify the origin of logarithmically enhanced contributions in the back-to-back limit as the exchange of operators with low twists and large spins in the local operator product expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Using techniques from the conformal bootstrap, the large logarithms beyond leading power can be resummed to all orders in the perturbative coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' As an illustration of this method, we perform an all-order re- summation of the leading and next-to-leading logarithms beyond the leading power in N = 4 Super Yang-Mills theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' INTRODUCTION Distributions of energy flows in high energy scattering in Quantum Chromodynamics (QCD) encode unique in- formation of the Lorentizian dynamics of quantum gauge theory, which is otherwise hard to extract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' A famous example is the formation of jets, which are collimated sprays of hadrons that manifest the underlying struc- ture of of quark and gluon scattering [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' From the early days of QCD, global energy flow distributions have been characterized using infrared and collinear safe shape functions [3–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Alternatively, it can also be studied us- ing statistical correlation of final-state energy flows, of which the simplest one is the two-point Energy-Energy Correlator (EEC) [12–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' In fact, the two different approaches for analyzing energy flow distributions are closely related by an integral transformation [16–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' In the e+e− scattering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' EEC can be written as the Fourier transformation of the correlation function of en- ergy flow operators EEC(y) = 8π2 q2σ0 � d4x eiq·x13⟨Jµ(x1)E(n2)E(n4)J† µ(x3)⟩ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (1) where n2 = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='⃗n2) and n4 = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='⃗n4) specify the direc- tions of the detected energy flow in the collider,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' y = 1 − (n2·n4)q2 2(n2·q)(n4·q) = (1 + cos θ)/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' q2 > 0 is a timelike momentum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Jµ is the electromagnetic current,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' and the energy flow operator is defined as a detector time integral of the energy-momentum tensor [20–27] E(ni) = ∞ � −∞ d ni·xi 16 lim ¯ni·xi→∞(¯ni · xi)2Tµν(xi)¯nµ i ¯nν i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (2) Experimental studies of EEC have a long history [28–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' They have been used to provide precision extraction of strong coupling constant [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Intuitively, when two narrow jets are produced in e+e−, they tend to be back-to-back due to momentum conservation, reflecting the underlying q¯q production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' This leads to a peak for EEC as y → 0, known as the Sudakov peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' In perturbation theory, it is manifested as large logarithmic corrections, EEC(y) ∼ ∞ � n=1 2n−1 � m=0 αn s � cn,m logm y y + dn,m logm y � , (3) where we have only shown the Leading Power (LP, ∼ y−1) and the Next-to-Leading Power (NLP, ∼ y0) cor- rections for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The leading power series of EEC resembles the perturbative structure of vector boson pro- duction at small pT and exhibits lightcone divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' It can be resummed to all orders in the perturbative cou- pling by solving a 2d renormalization group equation in virtuality and rapidity [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Using the recently avail- able 4-loop rapidity anomalous dimension [42, 43], its perturbative resummation has been performed to N4LL accuracy [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' However, power corrections to EEC, and in general to event shape functions and transverse mo- mentum dependent observables, are much less under- stood, both perturbatively and non-perturbatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Re- cently there have been significant developments towards a more satisfying picture of power corrections for various observables, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' [44–88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Yet the status is still far from that of leading power terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' In this work we initiate a study of power corrections to EEC by exploiting conformal symmetry and techniques from the analytic conformal bootstrap [89–92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='1 Building upon an important observation by Korchemsky [99], we connect the logarithmically enhanced terms in power cor- rections (m > 0 in (3)) with the expansion of correlators around the double lightcone limit, which is controlled by the twist expansion and the large spin expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Using 1 Applications of conformal symmetry in QCD has a long his- tory [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' For recent applications, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' [42, 43, 94–98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='03616v1 [hep-ph] 9 Jan 2023 2 t lightlike infinity x1 x3 x2 x4 r FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Penrose diagram for the double lightcone limit of the local correlator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' twist conformal blocks [100], tails from large spins can be systematically resummed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Further simplifications come from crossing symmetry, which relates the twist correc- tions with large spin corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Using this method, we explicitly carry out a calculation in N = 4 super Yang- Mills (SYM) theory, and achieve the first Leading and Next-to-Leading Logarithmic resummation at the sub- leading power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' BACK-TO-BACK V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' DOUBLE LIGHTCONE As is clear from (1), EEC is related to the local Wight- man correlator ⟨Ω|J(x1)T(x2)T(x4)J(x3)|Ω⟩ by detector time integrals and a Fourier transform [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' It is use- ful to understand which region of the local correlator cor- responds to the y → 0 limit of EEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Let us backtrack the dominant contribution by first undoing the Fourier trans- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' We can apply a Lorentz transformation to make the two detectors exactly back-to-back: nµ 4 → ¯nµ 2 = (1, −⃗n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' At the same time, the momentum qµ gains a small transverse component ⃗q⊥, which is schematically related to y as |⃗q⊥|2/q2 ∼ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' In position space, this cor- responds to the region where |(n2·x13)(¯n2·x13)/x2 13| ∼ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Since E(n2)E(¯n2) is invariant under the boost along ⃗n2, we have an extra degree of freedom to choose a frame such that n2 · x13, ¯n2 · x13 ∼ √y|x13|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' In such a frame, the lightcone singularities of 1 and 3 are very close on the path of detector time integrals of 2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Therefore, we expect the dominant contribution to come from the region where 2 and 4 are near the pinch of lightcone sin- gularities: x2 12, x2 23, x2 34, x2 14 ≪ x2 13, x2 24, which is the null square configuration [101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Instead, we can also choose the frame that n2·x13 ∼ y|x13|, ¯n2·x13 ∼ |x13|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Then the leading y → 0 dependence comes from 2 being near the lightcone limit of both 1 and 3, which is called the double lightcone limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' In this work we delineate this correspon- dence and systematically go beyond the leading power results in [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' It is well known that the lightcone limit is controlled by twist expansion [102–105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The most singular behavior in the 1, 2-channel as x2 12 → 0 is produced by operators with the lowest twist τ = ∆ − ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' However, any single operator in the 1, 2-channel cannot generate the lightcone singularity in the crossed channel with x2 13 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The x2 13 → 0 singularity can only be produced by an infinite sum of operators with different spins at a given twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Therefore, the double lightcone limit, or relatedly the back-to-back limit of EEC, is determined by the large spin asymptotics of low-twist operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' To be specific, we now consider EEC in N = 4 SYM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Comments for QCD will be given at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' We consider the correlator of scalar operators belonging to the stress tensor supermultiplet ⟨O(x1)O(x2)O(x3)O(x4)⟩dyn = 1 (2π)4 x4 13x4 24 (x6 12x2 34)6 F(u, v) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (4) Here we have only kept the dynamical part and sub- tracted the contribution of protected operators which are not perturbatively corrected, see Supplemental Material for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Conformal symmetry ensures that F is a function of the conformal cross ratios u = x2 12x2 34 x2 13x2 24 = z¯z , v = x2 23x2 14 x2 13x2 24 = (1 − z)(1 − ¯z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (5) Expanded in the small coupling a = g2Nc 4π2 , F(u, v) reads F(u, v) = ∞ � n=0 anF(n)(u, v) = F(0)(u, v) + u3 v Φ(u, v) , (6) The new function Φ(u, v) = � n≥1 anΦ(n)(u, v) packages all the coupling dependent information and is crossing symmetric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=', Φ(u, v) = Φ(v, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' It has been calculated to three loops in [106, 107].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The correlator admits the following superconformal block decomposition [108, 109]: F(u, v) = � ∆ � even ℓ aτ,ℓG∆+4,ℓ(u, v) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (7) Here the sums are over all superconformal primary oper- ators with dimension ∆, spin ℓ and OPE coefficient aτ,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The use of τ in the label foreshadows the twist expansion later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The 4d bosonic conformal blocks are given by [110] G∆,ℓ(u, v) = z¯z ¯z − z [k∆−ℓ−2(z)k∆+ℓ(¯z) − (z ↔ ¯z)] , (8) where kβ(x) = xβ/22F1( β 2 , β 2 , β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' They are eigenfunc- tions of the quadratic Casimir operator D2 = z2((1 − z)∂2 z − ∂z) + (d − 2)z¯z z − ¯z (1 − z)∂z + (z ↔ ¯z) (9) 3 with eigenvalues 1 2 (∆(∆ − 4) + ℓ(ℓ + 2)) [111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Super- conformal symmetry causes a shift ∆ → ∆ + 4 in the expansion (7), and we denote Gτ,ℓ(z, ¯z) = G∆+4,ℓ(u, v) for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Perturbative corrections, via conformal block decomposition, are encoded in the expansions of twists and OPE coefficients τ = τ0 + ∞ � n=1 anγ(n) τ0,ℓ , aτ,ℓ = ∞ � n=0 ana(n) τ0,ℓ , (10) where τ0 is the classical twist and � n≥1 anγ(n) τ0,ℓ = γτ,ℓ is the anomalous dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The anomalous dimensions enter via the expansion for conformal blocks: Gτ,ℓ = �∞ n=0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='γn τ,ℓ∂n τ0Gτ0,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Note that ∂n τ0Gτ0,ℓ contains at most logn u in the small u limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The double lightcone limit corresponds to u, v → 0, or equivalently, z → 0, ¯z → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Each conformal block in this limit is controlled by the twist G∆,ℓ(z, ¯z) ∼ zτ/2 log(1 − ¯z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The presence of logarithms partially demonstrates the effect of summing over infinitely many spinning operators because each conformal block contains infinitely many descendants in a given conformal family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' However, we will encounter more divergent pieces than a single logarithm, which require the large spin contribu- tion from infinitely many primary operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Examples include power divergences (1 − ¯z)m<0 and powers of log- arithms [log(1 − ¯z)]k≥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The latter arises from logk u of ∂k τ0Gτ0,ℓ in small u under crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' In the following, we refer to these as enhanced divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' To lighten the notations, we will denote the logarithms as log(x) = Lx from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' We also point out that crossing symmetry will play an important role in our computation of the power correc- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' As we will see, the u, v power corrections to Φ(u, v) contribute equally to the y power corrections in EEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' In particular, EEC at NLP corresponds to order u0v1 and u1v0 in Φ(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The former only requires twist-2 data to subleading order in the large spin limit (O(ℓ−2)), while the latter contains twist-4 contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Crossing symmetry equates these two contributions and therefore avoids the necessity of inputting the twist-4 information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' CFT data needed at different orders Power Corrections Perturbative Corrections twist large spin LL NLL LP 2 O(ℓ0) a(0) 2,ℓ , γ(1) 2,ℓ a(1) 2,ℓ , γ(2) 2,ℓ NLP 2 O(ℓ−2) a(0) 2,ℓ , γ(1) 2,ℓ a(1) 2,ℓ , γ(2) 2,ℓ 4 O(ℓ0) a(0) 4,ℓ , γ(1) 4,ℓ a(1) 4,ℓ , γ(2) 4,ℓ 2 At one loop the perturbative logarithms at NLP have at most k = 1, therefore are not fixed by large spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Before going into the technical details, we provide a brief overview for the next two sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' We will first use techniques from large spin perturbation theory to extract the enhanced divergences in v at twist-2 up to NLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' That is, we will obtain Φ(u, v) = p0(Lu, Lv)+p1(Lu, Lv)v+· · · , in which p0, p1 are polynomials in Lu, Lv at each pertur- bative order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Crossing symmetry then fixes the NLP con- tribution in u to be Φ(u, v) = p0(Lu, Lv)+p1(Lu, Lv)v + p1(Lv, Lu)u · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The relevant data is listed in Table I, but only the first two rows are needed thanks to crossing symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Finally, we map the small u, v expansion of Φ(u, v) to the back-to-back limit of EEC(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The rules at LP and NLP are given in Table II and are valid to NLL accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' We will explain these points in more detail in the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Logarithms Map at LP and NLP Φ(u, v) 4y(1 − y)2 × EEC(y) LP Lm u Lm v 2(m + n)Lm+n−1 y/(1−y) NLP uLm u Ln v 2m(1−m) m+n−1 y 1−y Lm+n−1 y/(1−y) vLm u Ln v 2n(1−n) m+n−1 y 1−y Lm+n−1 y/(1−y) LARGE SPIN ANALYSIS The enhanced divergences can be systematically han- dled by using the Large Spin Perturbation Theory [100], which culminates an array of earlier works in the large spin sector [112–118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The starting point is the free the- ory limit where the twists are degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The correlators can be written as a sum over the twists F(0)(z, ¯z) = � τ0=2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Hτ0(z, ¯z) , (11) where each Hτ0(z, ¯z) sums over spins Hτ0(z, ¯z) = ∞ � ℓ=0 ⟨a(0) τ0,ℓ⟩Gτ0,ℓ(z, ¯z) , (12) and is known as a twist conformal block (TCB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' When the interaction is turned on, the twist degeneracies are lifted and the OPE coefficients are corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' In general, we find that the expansion consists of sums of the form ∞ � ℓ=0 ⟨a(0) τ0,ℓ⟩κτ0(ℓ)Gτ0,ℓ(z, ¯z) , (13) where the quantities κτ0(ℓ) admit expansions around large conformal spins J2 τ,ℓ = (ℓ + τ 2)(ℓ + τ 2 − 1) with the following schematic form κτ0(ℓ) = ∞ � m=0 N � i=0 Km,i J2m ˜τ0,ℓ logi J2 ˜τ0,ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (14) 4 Here we introduce shifted twists ˜τ0 = τ0 + 4 to take into account the dimension shift in the decomposition (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' An interesting feature, as we will see in explicit calculations, is that only even negative powers appear in the expan- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' This is closely related to the reciprocity relation [119, 120], which has been explicitly verified in QCD to three loops [121].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Consequently, the correlator should now be expanded in terms of a more general class of TCBs H(m,i) τ0 (z, ¯z) = � ℓ a(0) τ0,ℓ logi J2 ˜τ0,ℓ J2m ˜τ0,ℓ Gτ0,ℓ(z, ¯z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (15) Note that conformal blocks satisfy the Casimir equation C˜τ0Gτ0,ℓ(z, ¯z) = J2 ˜τ0,ℓGτ0,ℓ(z, ¯z) , (16) where Cτ = D2 + 1 4τ(2d − τ − 2) is the shifted confor- mal Casimir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' It follows that TCBs obey the following recursion relations H(m,i) τ0 (z, ¯z) = C˜τ0H(m+1,i) τ0 (z, ¯z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (17) The full TCBs are in general difficult to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' How- ever, we will only need them in the small v limit where computations become manageable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The TCBs in 4D take a factorized form H(m,i) τ0 (z, ¯z) = z k˜τ0−2(z) ¯z − z ¯H(m,i) τ0 (¯z) , (18) where we have dropped the regular part when ¯z → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Moreover, the recursion relation (17) becomes ¯H(m,i) τ0 (¯z) = ¯D ¯H(m+1,i) τ0 (¯z) , (19) where ¯D = ¯z2(1 − ¯z) d2 d¯z2 − ¯z(2 − ¯z) d d¯z + 2 − ¯z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (20) Using this recursion relation, one can compute the TCBs explicitly in the small v limit [122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' For example, we find ¯H(0,i) 2 (¯z) = (−1)i �Li ϵ 2ϵ + Li+1 ϵ 6(i + 1) + γE − 3 3 Li ϵ � + · · · , ¯H(1,i) 2 (¯z) = (−1)i � Li+2 ϵ 2(i + 1)(i + 2) + γELi+1 ϵ i + 1 � + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (21) where we have defined ϵ = 1 − ¯z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Let us now focus on the small u limit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=', z → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Together with ¯z → 1, the correlator takes the form F(n) = z3 logn z ϵ � logn ϵ(An,1 + An,2ϵ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=') + logn−1 ϵ(Bn,1 + Bn,2ϵ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=') + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' � + O(z4) = z3 ∞ � m=0 n � i=0 Cm,i ¯H(m,i) 2 (¯z) + O(z4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' An important point is that to compute the NqLP in the small ϵ expansion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=', An,j=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=',q, Bn,j=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=',q etc, we only need ¯H(m,i) 2 with m = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' , q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' To see this, we act on the two expansions with ¯D and compare the power divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Note that for any polynomial p(ϵ) ¯D(p(ϵ) logi ϵ) = i(i − 1)(1 − ϵ)p(ϵ) logi−2 ϵ ϵ + O(ϵ0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Taking p(ϵ) = ϵq at the q-th order, we find the RHS only becomes a power divergence after acting q times with ¯D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' On the other hand, it is known that only the TCBs with m ≤ 0 are power divergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The repeated ¯D action makes the TCBs with m ≤ q power divergent and therefore responsible for the q-th order correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' We now explicitly compute the power corrections using the TCB decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' From Table I, to NLL and 2nd order in NLP the needed data is [122–124] a(0) 2,ℓ = Γ(ℓ + 3)2 Γ(2ℓ + 5) , (22) γ(1) 2,ℓ = log J2 6,ℓ + 2γE + 1 3J2 6,ℓ + O(J−4 6,ℓ ) , (23) a(1) 2,ℓ a(0) 2,ℓ = �1 + 4J6,ℓ 16J2 6,ℓ −log 2 � γ(1) 2,ℓ − ζ2+ 1 J6,ℓ +O(J−3 6,ℓ ),(24) γ(2) 2,ℓ = � 1 J6,ℓ − ζ2 2 � γ(1) 2,ℓ − 3ζ3 2 + 1 J2 6,ℓ + O(J−3 6,ℓ ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (25) From the conformal block decomposition (7), the small u expansion of F(n) up to NLL accuracy reads F(n) = z3 � even ℓ a(0) 2,ℓ �� γ(1) 2,ℓ �n 2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Ln z + � γ(1) 2,ℓ �n−1 Ln−1 z 2n−1(n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' × � a(1) 2,ℓ a(0) 2,ℓ + (n − 1) γ(2) 2,ℓ γ(1) 2,ℓ + γ(1) 2,ℓ ∂ℓ 2 � � k2ℓ+6(¯z) + · · · , (26) where all the odd powers in 1/J6,ℓ cancel out upon using the large spin expansion of CFT data (23-25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' We can therefore rewrite it in terms of TCBs as F(n) = (27) Ln z n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' � i �n i � �γn−i E 2i H(0,i) 2 + n − i 3 γn−1−i E 2i+1 H(1,i) 2 � + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' We have only showed the LL part for brevity and left the NLL part to Supplemental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Substituting in the TCBs using (18) and (21) we obtain F(n) at LP in z and NLP in 1 − ¯z F(n) = (−1)nz3 2n+1n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Ln z Ln ϵ �1 − ϵ ϵ + 2n 3Lϵ + 1 Lz +· · · � +O(z4) , (28) 5 with NLL accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Using F(n)(z, ¯z) = v u3 Φ(n)(u, v) and crossing symmetry Φ(u, v) = Φ(v, u), we get the NLL prediction for Φ(n) at NLP in the double lightcone limit Φ(n) = (−1)n 2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Ln uLn v �1 2 + (u + v) (29) + ��n + 1 2 u + n 3 v � 1 Lv + (u ↔ v) � + · · · � , n > 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' As was promised in the last section, only twist-2 CFT data was used in the whole process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' This prediction is checked against the available two- and three-loop results in the Supplemental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' POWER CORRECTIONS TO EEC IN N = 4 SYM The final task is to find the explicit relation between the double lightcone limit series uj1vj2Lm u Ln v and the back-to-back limit series yjLk y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The answer can be found using the Mellin representation of EEC [25, 26, 125] EEC(y) = 1 4y(1 − y)2 � dj1dj2 (2πi)2 M(j1, j2)K(j1, j2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' y) , (30) where M(j1, j2) is the Mellin amplitude for Φ(u, v): Φ(u, v) = � dj1dj2 (2πi)2 M(j1, j2)uj1vj2 and the kernel K(j1, j2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' y) is defined as K(j1, j2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' y) = 2Γ(1 − j1 − j2) � y 1−y �j1+j2 Γ(j1 + j2) [Γ(1 − j1)Γ(1 − j2)]2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (31) This gives the map uj1vj2 → K(j1,j2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='y) 4y(1−y)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Taking deriva- tives w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' j1, j2 generates the rules containing loga- rithms in u, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' One subtlety is that, due to the presence of the pole at j1 + j2 = 1 in K(j1, j2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' y), the maps at NLP cannot predict the y0 term without any log y en- hancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' But it can be shown that all the logarithmic contributions at NLP are preserved (see Supplemental Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The rules at LP and NLP up to NLL are summarized in Table II, and lead to EEC(n>1)(y) = (−1)n 2n(n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' � 1 2y � L2n−1 y + O(L2n−3 y ) � + � n 2n − 1L2n−1 y + 7n − 5 12 L2n−2 y + O(L2n−3 y ) � + · · · � , (32) which is in full agreement with the full theory calculation up to n = 3 in [126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The n > 3 terms are new and are one of the main result of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The analytic series in n can be resummed explicitly to all orders, leading to 160 165 170 175 180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='8 θ EEC(θ) EEC Back-to-Back Limit Resummation FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' EEC as a function of θ in the back-to-back limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' We use g2/(4π) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='118 to mimic the QCD strong coupling at Z pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The dashed line refers to LP resummed to NLL, with the inclusion of NLP terms up to NNLO (n ≤ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' the following NLL formula at LP and NLP3 EEC(y) = −aLye− aL2 y 2 4y − 1 4 ��π 2 √a erf ��a 2Ly � +aLye− aL2 y 2 � + a 48(7aL2 y − 4)e− aL2 y 2 + a 12 + · · · , (33) where erf is the error function erf(x) = 2 √π � x 0 e−t2dt 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' 2 we plot the N = 4 EEC in the back-to-back limit to illustrate the importance of NLP resummation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' It can be seen that the LL and NLL series at NLP leads to substantial corrections for not too large θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' For θ > 175◦ the Sudakov double logs suppressed the NLP contribu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' For comparison we also plot in dashed line the fixed-order NLP results truncated to NNLO [126], along with the LP NLL series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' In this case sizable NLP cor- rections can be found for θ > 175◦, which however is misleading as they disappear after resumming to all or- ders in coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' DISCUSSIONS Our results lead to several exciting research avenues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' First of all, it is interesting to apply the results to EEC in QCD, where fixed-order data up to NLO has become 3 The one-loop NLL contribution at NLP has no Ly enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Therefore, we need to input the one-loop EEC to fix the constant a/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' 4 We note that the LL-NLP series has been studied in [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Their results disagree with ours starting from O(a3), and seems to be in conflict with the fixed-order analytic result in [126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Further comparison is provided in the Supplemental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' 6 available recently [127–129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The local correlator of four electromagnetic currents in QCD has also been computed at one loop [130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Secondly, in QCD running coupling corrections will modify NLL series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' It would be impor- tant to understand how to incorporate these effects while retaining the power of conformal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Thirdly, our results provide concrete data for quantitative com- parison between additive and multiplicative scheme in resummation matching, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' [131].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Fourthly, local correlators exhibit other interesting limits, such as the Regge limit [132].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' It would be interesting to understand what constraints are imposed on EEC by such limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Last but not least, it would be worthwhile to under- stand the relation between our approach and the conven- tional approach based on momentum space renormaliza- tion group, in particular the relation between crossing symmetry for local correlator and the consistency rela- tions from infrared poles cancellations [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' We thank Zhongjie Huang, Kai Yan, and Xiaoyuan Zhang for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' are sup- ported by the Natural Science Foundation of China un- der contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' 11975200 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' 12147103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' is supported by funds from UCAS and KITS, and by the Fundamental Research Funds for the Central Universi- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' ∗ chenhao201224@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='cn † xinan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='zhou@ucas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='cn ‡ zhuhx@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='cn [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Hanson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Rev.' metadata={'source': 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Ebert, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Moult, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Stewart, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Tack- mann, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Vita, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} 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+page_content=' Eden, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Petkou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Schubert, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Sokatchev, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' B 607, 191 (2001), arXiv:hep-th/0009106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' 9 SUPPLEMENTAL MATERIAL Scalar Local Correlator To make precise the local correlator in (4), we consider the following operator made out of the six scalars φI=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=',6 of N = 4 SYM OIJ = tr � φIφJ� − 1 6δIJtr � φKφK� , (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='1) The operator is the superprimary of the stress tensor multiplet and transforms in the symmetric traceless represen- tation of the SO(6) R-symmetry group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' It is convenient to keep track of the R-symmetry information by contracting the indices with null polarization vectors YI or with a traceless symmetric tensor SIJ 5 O(x, Y ) ≡ OIJ(x)YIYJ , or O(x, S) = OIJ(x)SIJ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='2) Due to superconformal symmetry, the four-point function has following “partially non-renormalized” form [133] ⟨O(x1, Y1)O(x2, Y2)O(x3, Y3)O(x4, Y4)⟩ = G(0)(1, 2, 3, 4) + 2(N 2 c − 1) (4π2)4 y4 12y4 34 x2 12x2 34x2 14x2 23 R Φ(u, v) (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='3) where G(0)(1, 2, 3, 4) is the tree-level correlator G(0)(1, 2, 3, 4) =(N 2 c − 1)2 4(4π2)4 �� y2 12y2 34 x2 12x2 34 �2 + � y2 13y2 24 x2 13x2 24 �2 + � y2 41y2 23 x2 41x2 23 �2� +N 2 c − 1 (4π2)4 � y2 12y2 23y2 34y2 41 x2 12x2 34x2 23x2 41 + y2 12y2 24y2 34y2 13 x2 12x2 24x2 34x2 31 + y2 13y2 23y2 24y2 41 x2 13x2 23x2 24x2 41 � , (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='4) with y2 ij ≡ Yi · Yj and the function Φ(u, v) encodes all the dynamical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The factor R is determined by superconformal symmetry R = (1 − zα)(1 − z¯α)(1 − ¯zα)(1 − ¯z¯α) , (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='5) where the conformal cross ratios z, ¯z have already been introduced in the main text and α, ¯α are similarly the R-symmetry cross ratios defined by y2 13y2 24 y2 12y2 34 = α¯α , y2 14y2 23 y2 12y2 34 = (1 − α)(1 − ¯α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='6) The weak coupling g ≪ 1 expansion of Φ(u, v) reads Φ(u, v) = ∞ � n=1 anΦ(n)(u, v) , a = g2Nc 4π2 , (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='7) and is known up to three loops [107].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Since the dynamic function Φ(u, v) is R-symmetry independent, we can choose special polarizations to simplify the correlator (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Following [25], let us take YI = (1, 0, 1, 0, i, i) , SIJ = diag(1, −1, 0, 0, 0, 0) , S′ IJ = diag(0, 0, 1, −1, 0, 0) , (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='8) and define � O(x2) = 2O(x2, S) , � O′(x4) = 2O(x4, S′) , O(x3) = �N 2 c − 1 2π4 �− 1 2 O(x3, Y ) , O†(x1) = �N 2 c − 1 2π4 �− 1 2 O(x1, Y ∗) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='9) 5 The tensor can be built from the vectors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=', SIJ = Y ′ I Y ′ J + Y ′′ I Y ′′ J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Therefore, it is sufficient to focus on the former case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' 10 Then the four-point function becomes ⟨O†(x1) � O(x2) � O′(x4)O(x3)⟩ = 1 (2π)4 1 (x2 12x2 34)2 �N 2 c − 1 8 � 1 + u2 v2 � + u v �1 2 + Φ(u, v) �� = 1 (2π)4 1 (x2 12x2 34)2 �N 2 c − 1 8 Gshort(u, v) + 1 u2 F(u, v) � , (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='10) where we have further split it into short multiplet contribution Gshort(u, v) and the long multiplet contribution F(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The explicit form of Gshort(u, v) can be found in [109] and is protected from perturbative corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' As a result, Φ(u, v) is essentially the same as all the loop corrections (n ≥ 1) of F(u, v) = � n≥0 anF(n)(u, v), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=', F(u, v) − F(0)(u, v) = u3 v Φ(u, v) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='11) Another reason for choosing the polarizations (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='8) is the relation to the scalar detectors S(ni) = 1 4 ∞ � −∞ dni · xi lim ¯ni·xi→∞(¯ni · xi)2O(xi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='12) Thanks to superconformal symmetry, the spinning correlator ⟨JTTJ⟩ is related to the scalar correlator (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='3) by Ward identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Moreover, the EEC and the scalar-scalar correlation (SSC) are also proportional [25] ⟨E(n2)E(n4)⟩ = 4(q2)2 (n2 · n4)2 ⟨S(n2)S(n4)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='13) Twist Conformal Blocks The TCBs with logarithms can be computed using the method of [122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The idea is to apply the recursion relations on ¯H(0,0) τ0 (¯z), which is determined by the tree-level correlator, and compute ¯H(m,0) τ0 (¯z) at negative integer m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' We then analytic continue in m and take derivatives to obtain the logarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' For our case, τ0 = 2 and we have ¯H(0,0) 2 (¯z) = ¯z2(2 − ¯z) 2(1 − ¯z) + ¯z log(1 − ¯z) = 1 2ϵ + regular terms .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='14) Repeated ¯D action gives ¯H(m,0) 2 (¯z) and the first few terms in small ϵ for negative integer m are given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='37) of [122] ¯H(m,0) 2 (¯z) =1 2ϵm−1Γ(1 − m)2 + 1 6m � 2m2 − 6m + 1 � ϵmΓ(−m)2 + 1 180(m − 1)m(m + 1) � 20m3 − 54m2 − 35m + 36 � ϵm+1Γ(−m − 1)2 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='15) Obtaining the full analytic expression for ¯H(m,i) 2 (ϵ) is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' But life is much easier if we content ourselves with getting first few orders in ϵ and logarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Truncated to order ϵ0, only ¯H(0,i) 2 (ϵ) and ¯H(1,i) 2 (ϵ) are relevant and we find ¯H(0,i) 2 (¯z) = (−1)i ϵ �1 2 logi ϵ + iγELi−1 ϵ + i(i − 1) 12 (12γ2 E + π2)Li−2 ϵ + · · · � (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='16) + (−1)i � 1 6(i + 1)Li+1 ϵ + γE − 3 3 Li ϵ + π2 + 12γ2 E − 72γE + 12 36 iLi−1 ϵ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' � + · · · , ¯H(1,i) 2 (¯z) = (−1)i � 1 2(i + 1)(i + 2)Li+2 ϵ + γE i + 1Li+1 ϵ + 12γ2 E + π2 12 Li ϵ + · · · � + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='17) 11 More Details of (26) and (27) In this section, we present the details of the large spin perturbation calculation needed for obtaining EEC in the back-to-back limit to the NLL and NLP order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' As we explained in the main text, only twist-2 contributions are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The expansion of the twist-2 conformal block G2+γ2,ℓ,ℓ(z, ¯z) in the z → 0 limit is G2+γ2,ℓ,ℓ(z, ¯z) = z3+γ2,ℓ/2k6+2ℓ+γ2,ℓ(¯z) + O(z4) = z3 ∞ � n=0 an � � 1 2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' � γ(1) 2,ℓ �n logn z + 1 2n−1(n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='γ(2) 2,ℓ � γ(1) 2,ℓ �n−2 logn−1 z + · · · � k6+2ℓ(¯z) + 1 2n(n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' � γ(1) 2,ℓ �n logn−1 z ∂ℓk6+2ℓ(¯z) + · · · � + O(z4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='18) Combined with the expansion of the OPE coefficient a2,ℓ, we obtain the leading twist contribution to F(z, ¯z) F(n)(z, ¯z) = z3 � even ℓ � logn z 1 2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='a(0) 2,ℓ � γ(1) 2,ℓ �n k6+2ℓ(¯z) + logn−1 z � a(1) 2,ℓ � γ(1) 2,ℓ �n−1 2n−1(n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' + a(0) 2,ℓ γ(2) 2,ℓ � γ(1) 2,ℓ �n−2 2n−1(n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' � k6+2ℓ(¯z) + logn−1 z a(0) 2,ℓ 1 2n(n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' � γ(1) 2,ℓ �n ∂ℓk6+2ℓ(¯z) + · · · � + O(z4) , (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='19) which is NLL in log z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Then using the IBP identity a(0) 2,ℓ∂ℓk6+2ℓ(¯z) = ∂ℓ[a(0) 2,ℓk6+2ℓ(¯z)] − k6+2ℓ(¯z)∂ℓa(0) 2,ℓ , (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='20) and a(1) 2,ℓ = −ζ2a(0) 2,ℓ + 1 2∂ℓ � a(0) 2,ℓγ(1) 2,ℓ � , we rewrite (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='19) as F(n)(z, ¯z) =z3 � even ℓ � 1 2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' logn z a(0) 2,ℓ � γ(1) 2,ℓ �n k6+2ℓ(¯z) + 1 2n(n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' logn−1 z � � γ(1) 2,ℓ �n ∂ℓ � a(0) 2,ℓk6+2ℓ(¯z) � +a(0) 2,ℓk6+2ℓ(¯z) � γ(1) 2,ℓ �n−2 �� −2ζ2 + ∂ℓγ(1) 2,ℓ � γ(1) 2,ℓ + 2(n − 1)γ(2) 2,ℓ � � + · · · � + O(z4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='21) The use of (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='20) becomes clear when we use the integer-step finite difference to approximate ∂ℓ � a(0) 2,ℓk6+2ℓ(¯z) � at large spin ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' On a general function f(ℓ), we approximate f ′(ℓ) ≈ f(ℓ+2)−f(ℓ−2) 4 , which is accurate up to O(ℓ−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Neglecting boundary terms which vanish at large spins, we can write � even ℓ � γ(1) 2,ℓ �n ∂ℓ � a(0) 2,ℓk6+2ℓ(¯z) � ≈ 1 4 � even ℓ a(0) 2,ℓk6+2ℓ(¯z) �� γ(1) 2,ℓ−2 �n − � γ(1) 2,ℓ+2 �n� , (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='22) Expanding everything other than a(0) 2,ℓ with respect to the large conformal spin J2 6,ℓ, we get F(n)(z, ¯z) = z3 � even ℓ � 1 2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' logn z a(0) 2,ℓk6+2ℓ(¯z) � � log(J2 6,ℓ) + 2γE �n + n 3 � log(J2 6,ℓ) + 2γE �n−1 1 J2 6,ℓ + · · · � + 1 2n(n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' logn−1 z a(0) 2,ℓk6+2ℓ(¯z) � − (n + 1)ζ2 � log(J2 6,ℓ) + 2γE �n−1 − 3(n − 1)ζ3 � log(J2 6,ℓ) + 2γE �n−2 + 1 J2 6,ℓ � (n − 1) � 1 − ζ2 3 (n + 1) � � log(J2 6,ℓ) + 2γE �n−2 −(n − 1)(n − 2)ζ3 � log(J2 6,ℓ) + 2γE �n−3 �� + · · · � + O(z4) , (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='23) 12 which can be organized into TCBs as F(n)(z, ¯z) =logn z 2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' � n � i=0 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (n − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (2γE)iH(0,n−i) 2 + n 3 n−1−i � i=0 (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (n − 1 − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (2γE)iH(1,n−1−i) 2 + · · · � + logn−1 z 2n(n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' � − (n + 1)ζ2 n−1 � i=0 (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (n − 1 − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (2γE)iH(0,n−1−i) 2 − 3(n − 1)ζ3 n−2 � i=0 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (n − 2 − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (2γE)iH(0,n−2−i) 2 + (n − 1) � 1 − ζ2 3 (n + 1) � n−2 � i=0 (n − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (n − 2 − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (2γE)iH(1,n−2−i) 2 − (n − 1)(n − 2)ζ3 n−3 � i=0 (n − 3)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (n − 3 − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (2γE)iH(1,n−3−i) 2 + · · · � + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='24) Substituting the explicit TCBs (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='16, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='17), we get F(n)(z, ¯z)= z3 � 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' logn z �1 ϵ �(−1)n 2n+1 logn ϵ + · · · � + �(−1)n+1 2n+1 logn ϵ + (−1)nn 3 × 2n logn−1 ϵ + · · · � + · · · � +logn−1 z (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' �1 ϵ �(−1)n 2n+1 (n + 1)ζ2 logn−1 ϵ − (−1)n 2n+1 3(n − 1)ζ3 logn−2 ϵ + · · · � + �(−1)n 2n+1n logn ϵ + (−1)n+1 2n+1 (n + 1)ζ2 logn−1 ϵ + · · · � � + · · · � + O(z4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='25) Via F(n)(z, ¯z) = v u3 Φ(n)(z, ¯z), this gives the expansion of Φ(n)(z, ¯z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Crossing symmetry allows us to further recon- struct the O(u1v0) contributions, which gives the results in (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Details of the Map from uj1vj2Lm u Ln v to yjLk y In this section, we provide more details for establishing the map from the small u, v expansion of Φ(u, v) to the small y expansion of EEC(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Instead of electromagnetic current sources Jµ, we consider two scalar operator sources, belonging to the stress tensor multiplet, in the center of mass frame qµ = (Q, 0, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' EEC(y) relates to ⟨E(n2)E(n4)⟩ by an overall factor: EEC(y) = � dΩ2dΩ4δ(⃗n2 · ⃗n4 − cos θ)⟨E(n2)E(n4)⟩ Q2 = 8π2 Q2 ⟨E(n2)E(n4)⟩ , (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='26) where θ is the angle between ⃗n2 and ⃗n4 and we assume the convention that ⟨E(n2)E(n4)⟩ has already been normalized to the cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The superconformal Ward identities further reduce the EEC ⟨E(n2)E(n4)⟩ to scalar-scalar correlation (SSC) ⟨S(n2)S(n4)⟩ [25, 125] ⟨E(n2)E(n4)⟩ = 4(q2)2 (n2 · n4)2 ⟨S(n2)S(n4)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='27) The SSC is related to the local correlator in a simple way in Mellin space [25] Φ(u, v) = � dj1dj2 (2πi)2 M(j1, j2)uj1vj2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='28) Here M(j1, j2) is the Mellin amplitude and encodes all the dynamical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' To compute the SSC, the first step is to obtain the Lorentzian correlator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' This is achieved by using the Wightman prescription x2 ij → −x2 ij + iϵtij, if operator i sits before j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' In our case, the operator ordering is 1 < 2 < 4 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The second step is to perform the light transform on the Lorentzian correlator � i=2,4 � ∞ −∞ d(ni · xi) lim ¯ni·xi→∞ � ¯ni · xi 2 �2 , (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='29) 13 which turns local operators into detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' To obtain SSC in the momentum space, the last step is the Fourier transformation � d4x13 exp(iq · x13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' After normalized to the total cross section, the Mellin representation for SSC is ⟨S(n2)S(n4)⟩ = 1 (2π)4 π2 (n2 · n4)q2 1 − y y � dj1dj2 (2πi)2 M(j1, j2)K(j1, j2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' y) , (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='30) with K(j1, j2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' y) = 2Γ(1 − j1 − j2) Γ(j1 + j2) [Γ(1 − j1)Γ(1 − j2)]2 � y 1 − y �j1+j2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='31) Then using n2 · n4 = 2(1 − y) in the center of mass frame, we obtain the Mellin representation for EEC(y) EEC(y) = 8π2Q2 (1 − y)2 ⟨S(n2)S(n4)⟩ = 1 4y(1 − y)2 � dj1dj2 (2πi)2 M(j1, j2)K(j1, j2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='32) Therefore, we expect the following mapping from the double lightcone limit to back-to-back limit uj1vj2 → K(j1, j2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' y) 4y(1 − y)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='33) At LP and NLP, we need the following rules containing logarithms in u, v logm u logn v → m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' � |j1|=ϵ dj1 2πi 1 jm+1 1 � |j2|=ϵ dj2 2πi 1 jn+1 2 K(j1, j2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' y) 4y(1 − y)2 , (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='34) u logm u logn v → m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' � |j1−1|=ϵ dj1 2πi 1 (j1 − 1)m+1 � |j2|=ϵ dj2 2πi 1 jn+1 2 K(j1, j2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' y) 4y(1 − y)2 , (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='35) v logm u logn v → m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' � |j1|=ϵ dj1 2πi 1 jm+1 1 � |j2−1|=ϵ dj2 2πi 1 (j2 − 1)n+1 K(j1, j2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' y) 4y(1 − y)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='36) For the cases we will inspect, the only exception is the constant term at NLP which is caused by the j1 + j2 = 1 pole in K(j1, j2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' To see this, we consider the first derivative of 1−y y K(j1, j2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' y) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' y 1−y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Such an action shifts Γ(1−j1 −j2) to Γ(2−j1 −j2), which is analytic at j1 +j2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The constant term is killed after taking the derivative, while the logarithmic information remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The explicit expressions for general m, n up to NLL accuracy are shown in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Comparison with Existing Results In this section we provide a comparison of our predictions with the existing results in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' We begin with the local correlator in the double lightcone limit to NLL accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The full three-loop correlator can be found in [107]6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Upon expanding their results in the double lightcone limit we find Φ(1)(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' v) = � −1 4 log u log v + 0 · log(uv) + · · · � − �1 4(u + v) log u log v + 1 2(u log u + v log v) + · · · � + · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Φ(2)(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' v) = � 1 16 log2 u log2 v + 0 · log u log v log(uv) + · · · � + �1 8(u + v) log2 u log2 v + 3 16 log u log v(u log u + v log v) + 1 8 log u log v(v log u + u log v) + · · · � + · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Φ(3)(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' v) = � − 1 96 log3 u log3 v + 0 · log2 u log2 v log(uv) + · · · � − � 1 48(u + v) log3 u log3 v + 1 24 log2 u log2 v(u log u + v log v) + 1 48 log2 u log2 v(v log u + u log v) + · · · � + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='37) 6 There is an overall normalization difference � − 1 4π �n at the n-th loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' 14 Compared with our NLL prediction (29), we find perfect agreement except for the NLP terms at one loop and a single NLL-NLP term at two loops (shown in red explicitly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Such terms do not correspond to enhanced divergence in v and hence cannot be captured by the large spin analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Moreover, the two-loop NLL-NLP mismatched term 1 8 log u log v(v log u + u log v) does not map to NLL-NLP term in EEC, as can be checked from Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' For EEC in N = 4 SYM results up to three loops with full y dependence have been computed in [126], using the local correlator from [107] as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The back-to-back expansion of [126] up to NLL at NLP is EEC(1) = − 1 4y log y−1 2 log y+0 · y0 + O(y) , EEC(2) = 1 y �log3 y 8 + 0 · log2 y + · · · � + �log3 y 6 + 3 16 log2 y + · · · � + O(y) EEC(3) = 1 y � −log5 y 32 + 0 · log4 y + · · · � + � −3 log5 y 80 − log4 y 12 + · · · � + O(y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='38) To obtain (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='38), it is necessary to expand the following integral to NLP, E = � 1 0 d¯z � ¯z 0 dt −1 t(1 − y − ¯z) + y¯z � z¯z 1 − z − ¯z P1 + z2¯z (1 − z)2(1 − z¯z)P2 � , (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='39) where z = (1−y)t(t−¯z)/(t(1−y−¯z)+y¯z) and P1 and P2 are lengthy combinations of weight-3 harmonic polylogarithms and can be found in [126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The expansion of this integral begins from NLP and reads E = log5 y 60 + 0 · log4 y + 1 12ζ2 log3 y + · · · (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='40) where we have neglected terms of O(log2 y) and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' The expansion of this integral was also performed in [69], but the result reported in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='17) of [69] is larger than (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='40) by a factor of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Using (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='40) and expanding the remaining results in [126] with the package HPL, we obtain EEC(3) presented in (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Our results in (32) for n > 1 are in full agreement with EEC(2) and EEC(3) in (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' This provides a strong check for our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' Our large spin analysis does not expect to capture the NLP terms at one loop (shown in red in (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='38)), for the same reason as explained for local correlator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' We note that a LL study at NLP has also been performed in [69], where an RG equation at NLP has been derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' After changing to our normalization, their predicted LL coefficient at NLP and three loop, given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} +page_content='21) of [69], is −1/30 log5 y, which disagrees with our results in (33), as well as with the full results from [126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNE2T4oBgHgl3EQfCwYa/content/2301.03616v1.pdf'} diff --git a/UNFLT4oBgHgl3EQfQy8p/vector_store/index.faiss b/UNFLT4oBgHgl3EQfQy8p/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..2b81ccdefcd271c90aa82fc53a8316787b95ecbf --- /dev/null +++ b/UNFLT4oBgHgl3EQfQy8p/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:db34644ce3a82ecd0349b3454007765d76d675be18bc61e67cb0f90c224f4ec2 +size 8323117 diff --git a/V9E1T4oBgHgl3EQfJAMR/content/tmp_files/2301.02945v1.pdf.txt b/V9E1T4oBgHgl3EQfJAMR/content/tmp_files/2301.02945v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..adbcc37f075535fc51283c4f11591e74665ec11d --- /dev/null +++ b/V9E1T4oBgHgl3EQfJAMR/content/tmp_files/2301.02945v1.pdf.txt @@ -0,0 +1,929 @@ +Energy conversion efficiency from a high +order soliton to fundamental solitons in +presence of Raman scattering +ROBI KORMOKAR,* MD HOSNE MOBAROK SHAMIM, AND MARTIN ROCHETTE +Department of Electrical and Computer Engineering, McGill University, 3480 University Street, +Montréal, Québec, H3A 0E9, Canada +*robi.kormokar@mail.mcgill.ca +Abstract: We formulate the energy conversion efficiency from a high-order soliton to +fundamental solitons by including the influence of interpulse Raman scattering in the fission +process. The proposed analytical formula agrees closely with numerical results of the +generalized nonlinear Schrödinger equation as well as to experimental results, while the +resulting formulation significantly alters the energy conversion efficiency predicted by the +Raman-independent inverse scattering method. We also calculate the energy conversion +efficiency in materials of different Raman gain profiles such as silica, ZBLAN and +chalcogenide glasses (As2S3 and As2Se3). It is predicted that ZBLAN glass leads to the largest +energy conversion efficiency of all four materials. The energy conversion efficiency is a notion +of utmost practical interest for the design of wavelength converters and supercontinuum +generation systems based on the dynamics of soliton self-frequency shift. + +1. Introduction +Soliton self-frequency shift (SSFS) is an intrapulse Raman scattering process that is key for +wavelength conversion and supercontinuum generation [1-7]. Among the several nonlinear +wavelength conversion mechanisms, SSFS is of particular interest because of its wide tunability +range beyond tens of THz [8-11], and the absence of phase-matching constraints. SSFS is best +observed with short pulses, as SSFS increases proportionally with the fourth power of inverse +pulse duration [2, 12]. This is particularly the case when an Nth order soliton or mother soliton +(MS) experiences fission and splits into N fundamental solitons, from which solitons of +femtosecond duration typically emerge [1, 13]. Exploiting the soliton fission mechanism for +SSFS based wavelength conversion has been reported by different groups [14-16]. For +example, Gauthier et al. reported the soliton fission of a MS into solitons experiencing SSFS +for wavelength conversion up to 4.8 m in InF3 fiber [14]. Alamgir et al. reported continuously +tunable Raman solitons over the frequency range of 2.047-2.667 m using soliton fission and +SSFS in an As2S3 microwire [16]. Soliton fission is also a fundamental mechanism for +supercontinuum generation in fiber with anomalous dispersion [5]. The energy conversion +efficiency (ECE) from the MS to each fundamental soliton is of practical interest for +wavelength conversion applications. For instance, one may wish to optimize the energy transfer +from the MS specifically towards the fundamental soliton that experiences the largest amount +of SSFS [15, 16]. +Using the inverse scattering method (ISM), Kodama et al. have shown that an Nth order +soliton is a bound state of N fundamental solitons, as a solution to the nonlinear Schrödinger +equation (NLSE) [13]. They have also shown analytically that the bound state Nth order soliton +is breaking, i.e., experiences fission, under sufficient high-order linear and/or nonlinear +perturbation in the fiber. According to the ISM, each fundamental soliton after fission carries +an energy given by [1, 13, 17] + + +Ek = +(2N + 1 − 2k) +N2 +E0, +(1) + +where N = (0E0T0/2|2|)1/2 with E0 and T0 being the energy and duration of the MS, +respectively, 2 is the group velocity dispersion (GVD) coefficient, and 0 is the waveguide +nonlinear parameter at center frequency 0. The index k = 1…N labels each soliton emerging +from fission. Each soliton thus carries an energy Ek given by Eq. 1, leading to an ECE given by +ECEISM,k = (100Ek/E0)%. Of particular interest for wavelength conversion is the k = 1 Raman +soliton because it is the shortest, the most powerful, and most importantly, it experiences the +largest amount of SSFS. We however recently noticed numerically and experimentally that the +k = 1 Raman soliton emerging from soliton fission often carries significantly more energy than +predicted by the ISM, leading to ECE > ECEISM,1 [15]. +In this paper, we demonstrate that the ECE evaluated from numerical simulations and from +experiment agree together but diverge from the theoretical prediction of ECEISM,1. We +subsequently propose an improved analytical formulation of ECE for the k = 1 Raman soliton, +derived by adding an energy transfer contribution from interpulse Raman scattering in between +the k = 2…N solitons to the k = 1 soliton. We show that the ECEISM,1 diverges rapidly from +numerical predictions and experiment with MS of increasing order and decreasing duration. +This study clarifies the dynamics of Raman induced soliton fission, as well as providing a more +precise evaluation of ECE, a topic of utmost interest for the design of SSFS based wavelength +converters and supercontinuum sources. +2. Theory of Enhanced Energy Conversion Efficiency +2.1 Generalized Nonlinear Schrödinger Equation +The propagation of an optical pulse in a nonlinear and dispersive optical fiber is well described +by the generalized nonlinear Schrödinger equation (GNLSE) [1] + +∂A +∂z + 1 +2 (α0 + ∑ inαn +n! +∂n +∂Tn +∞ +n=1 +) A − i ∑ inβn +n! +∂nA +∂Tn +∞ +n=2 += +i (γ0 + ∑ inγn +n! +∂n +∂Tn +∞ +n=1 +) (A(z, T) ∫ R(T′)|A(z, T − T′)|2dT′ +∞ +0 + ), +(2) + +where A is the slowly varying pulse envelope, n = dn/dn, n = dn/dn, n = dn/dn are the +Taylor coefficients of the frequency dependent fiber loss (), propagation constant () and +nonlinear parameter (), respectively, evaluated at the center frequency 0. T is time in a frame +of reference moving at group velocity of the pulse. R(T) is the nonlinear response function +defined as [1] + +R(T) = (1 − fR)δ(T) + fRhR(T), +(3) + +where fR represents the fractional contribution of the delayed Raman response to nonlinear +polarization, (T) is the Dirac delta function, and hR(T) is the Raman response function. Using +the single peak Lorentzian model, hR(T) can be written as + +hR(T) = (τ1 +−2 + τ2 +−2)τ1 exp(− T τ2 +⁄ +) sin(T τ1 +⁄ +), +(4) + +where R = 1/1 is the angular frequency of the Raman gain peak and R = 1/2 is the bandwidth +of the Raman gain response [18]. + +2.2 Soliton Constitution and Fission +By ignoring high-order linear and nonlinear terms and by considering a lossless medium, the +NLSE can be written in a normalized form + +∂q +∂Z = i +2 +∂2q +∂τ2 + i|q|2q, +(5) + +with + +q = N +√P0 +A, +Z = z +LD +, +and τ = T +T0 +, +(6) + +where P0 and T0 are the pulse peak power and duration, and LD = T0 +2/|2| is the dispersion length. +By using the inverse scattering method developed by Zakharov et al. [19], Satsuma et al. +showed that a solution of Eq. (5) is an Nth order soliton of the form q = Nsech(), which itself +contains N fundamental solitons with amplitude [20] + +ηk = 2(N − k) + 1, k = 1, 2, … N +(7) + +and where fundamental solitons are expressed as [17] + +qk(τ, Z) = ηk sech[ηk(τ + κkZ − τ0k)] exp [−iκkτ + i +2 (ηk +2 − κk +2)Z − iσ0k], +(8) + +where k is related with the change in group velocity of the kth soliton component. Satsuma et +al. showed that k = 0 and all fundamental solitons of a high-order soliton travel at the same +group velocity, leading to a bound state of solitons [20]. Because of the phase interference +among fundamental solitons, the resulting high-order soliton profile and spectrum evolve +periodically with a period of Z = /2. By using Eq. (6) and (7) in Eq. (8) and by setting the +constant phase term 0k and the temporal offset of each soliton 0k to zero, the fundamental +solitons are expressed as + +Ak(T, 0) = +(2N + 1 − 2k) +N +√P0 sech [ +T +T0 (2N + 1 − 2k) +⁄ +]. +(9) + +From Eq. (9), the peak power and duration of each fundamental soliton component is + +Pk = +(2N + 1 − 2k)2 +N2 +P0, +(10a) + +and + +Tk = +T0 +(2N + 1 − 2k). +(10b) + +A high-order soliton becomes however unstable and eventually fissions under the influence +of linear and/or nonlinear perturbations such as Raman effect, third-order dispersion, and self- +steepening. As a result, fundamental solitons come apart and propagate with a peak power and +pulse duration defined in Eq. (10). According to the ISM, the ECE of each soliton is given by + + +ECEISM,k = +(2N + 1 − 2k) +N2 +× 100%. +(11) + +In a glassy medium such as an optical fiber, soliton propagation is also accompanied with +intrapulse Raman scattering that gradually transfers energy from short wavelength contents of +the pulse to the long wavelengths [2, 12]. As a result, the soliton experiences SSFS following +[2] + +Ω(z) = − 8TR|β2| +15T0 +4 zeff, +(12) + +where Ω is SSFS in rad/s, TR is Raman time, and + +zeff = 1 +4α (1 − e−4αz), +(13) + +is the effective length of the fiber. Equation (12) shows that SSFS increases proportionally with +the effective fiber length and with inverse pulse duration to the fourth order. Since the k = 1 +soliton has the shortest duration, according to Eq. (10b), this soliton therefore experiences the +largest amount of SSFS compared to the other soliton components. +Figure 1 shows a typical example of Raman induced soliton fission process of a N = 3 MS in +a 15 m long lossless standard single mode fiber (SSMF). To demonstrate the concept of Raman +induced soliton fission, the simulation only considers the Raman effect as the sole high-order +nonlinear effect. Fiber parameters are 2 = - 21.61 ps2/km, and 0 = 1.42 W-1/km at the initial +pulse wavelength of 0 = 1.55 m. The Raman response function given in Eq. (4) is used for +silica with 1 = 12.2 fs, 2 = 32 fs and fR = 0.18. The initial pulse duration and peak power are +283 fs and 1.7 kW, respectively. The simulation is performed by numerically solving the +GNLSE of Eq. (2) using a split-step Fourier method (details in section 4). As seen in Fig. 1, the +k = 1 Raman soliton emerges due to Raman induced soliton fission and experiences the most +SSFS. It is also observed that ECEk=1 = 66.0% whereas ECEISM,1 = 55.6%. The numerical +simulations and ISM results thus significantly differ in this example. + + +Fig. 1. Simulated (a) temporal and (b) spectral evolution of a third-order MS following a Raman induced soliton fission +process. The pulse initially has a center wavelength of 0 = 1.55 m, pulse duration of 283 fs and peak power of 1.7 kW. +The medium is a 15 m long lossless SSMF with 2 = - 21.61 ps2/km, 0 = 1.42 W-1/km at 0 = 1.55 m. +The observed ECE enhancement with respect to the ISM prediction is explained from a +transfer of energy in between solitons following the fission process, known as interpulse Raman +scattering [21, 22], and in contrast with intrapulse Raman scattering. In a bound state, a high- +order soliton experiences a periodic evolution due to a mutual interaction between GVD and +self-phase modulation (SPM) [17, 20]. High-order linear and/or nonlinear perturbations break + +1.2 +(a) +(b) +Input +Input +4 +Output +Output +0.8 +k = 1 soliton +0.6 +2 +0.4 +k = 1 soliton +人 +0.2 +01 +5 +10 +1.5 +1.55 +1.6 +1.65 +1.7down the bound state, which triggers a fission process, and the MS breaks into fundamental +soliton components. In any glassy medium, Raman scattering is the dominant mechanism that +triggers soliton fission [1]. The Raman effect transfers soliton energy towards lower frequency +components via intra- and interpulse Raman scattering. Intrapulse Raman scattering is +responsible for SSFS. Interpulse Raman scattering also occurs after the fission event, +amplifying solitons at long wavelength from solitons at short wavelengths, leading to energy +transfer in between solitons that temporally overlap. Since the k = 1 Raman soliton experiences +the most redshift from SSFS, it becomes amplified by the remaining k = 2…N solitons for as +long as they temporally overlap. For this reason, the k = 1 Raman soliton is expected to get an +enhanced ECE with respect to the ISM prediction. Note that for a MS with N being a non- +integer value, soliton fission is followed by fundamental solitons and dispersive waves. + +3. Analytical Formulation of Enhanced ECE +In addition to the ISM prediction of ECE that occurs during the fission of the MS, a new +formulation of ECE should also consider the energy transfer (ΔE) from the k = 2…N solitons +to the k = 1 Raman soliton resulting from interpulse Raman scattering that occurs moments +after fission. Taking interpulse Raman scattering into account, the new ECE of the k = 1 soliton +is expressed as + +ECEnew,1 = ECEISM,1 + Δe, +(14) + +where Δe is a correction term to the ECEISM,1. For interpulse Raman scattering to occur, the +temporal overlap in between involved pulses is required. An estimated pump energy from each +k = 2…N solitons to the k = 1 Raman soliton is expressed as + +Epk(z) = ∫ +Pk sech2 ( T +Tk +) dT +3T1+q +−3T1+q +, +(15) + +where T1 and q are the pulse duration and temporal position of the k = 1 Raman soliton, +respectively. Pk and Tk are the peak power and pulse duration of the k = 2…N solitons. The +integration limits in Eq. (15) are set such that when q = 0 the integration provides the energies +of the k = 2…N solitons that are overlapped with 99.0% of the total energy of the k = 1 soliton. +From Eq. (10) and integration, Eq. (15) transforms into + +Epk(z) = E0(2N + 1 − 2k) +N2 +tanh [3(2N + 1 − 2k) +(2N − 1) +] Λk(q), +(16a) + +where + +Λk(q) = +(1 − tanh2 q +Tk) +(1 − tanh2 3T1 +Tk tanh2 q +Tk) +. +(16b) + +The temporal position of the k = 1 soliton is evaluated from the moment method [23] + +q(z) = 4TR|β2|2 +15T1 +4 +z2. +(17) + +Therefore, the total pump energy from the k = 2…N soliton components sums up into + + +EpT(z) = ∑ E0(2N + 1 − 2k) +N2 +tanh [3(2N + 1 − 2k) +(2N − 1) +] Λk(q) +N +k=2 +. +(18) + +The ECE enhancement rate with propagation distance is proportional to EpT (z). Therefore, + +de +𝑑𝑧 ∝ ∑ ( +(2N + 1 − 2k) +N2 +tanh [3(2N + 1 − 2k) +(2N − 1) +]) Λk(q) +N +k=2 +. +(19) + +This rate also depends on the effective Raman gain experienced by the k = 1 Raman soliton, +quantified as + +de +𝑑𝑧 ∝ γ0P1fR ∫ +Im[h̃R(ω)] sech2 [π(ω − Ω(z))T0 +2(2N − 1) +] dω +0 +−∞ +, +(20) + +where P1 is the peak power of the k = 1 Raman soliton, and Im[h̃R(ω)] is the Raman gain +function [1]. (z) is the SSFS experienced by the k = 1 Raman soliton. By combining Eqs. (19) +and (20), the ECE enhancement rate becomes + +de +𝑑𝑧 ∝ ∑ ( +(2N + 1 − 2k) +N2 +tanh [3(2N + 1 − 2k) +(2N − 1) +]) Λk(q) +N +k=2 + +× γ0P1fR ∫ +Im[h̃R(ω)] sech2 [π(ω − Ω(z))T0 +2(2N − 1) +] dω +0 +−∞ +. +(21) + +By integration of Eq. (21) with respect to z, the correction term in Eq. (14) becomes + +Δe ∝ ∑ Ik ( +(2N + 1 − 2k) +N2 +tanh [3(2N + 1 − 2k) +(2N − 1) +]) +N +k=2 +, +(22) + +where + +Ik(z) = ∫ Λk(q)γ0P1fR ∫ +[Im[h̃R(ω)] sech2 [π(ω − Ω(z))T0 +2(2N − 1) +]] dω +0 +−∞ +dz +z +0 +. +(23) + +From Eq. (14) and Eq. (22), the correction to the ECE predicted by the ISM becomes + +ECEnew,1 = ECEISM,1 [1 + κ ∑ Ik +(2N + 1 − 2k) +(2N − 1) +tanh (3(2N + 1 − 2k) +2N − 1 +) +N +k=2 +], +(24) + +including a material-dependent proportionality constant + +κ = +10 +∫ +Im[h̃R(ω)]dω +0 +−∞ + +(25) + +where the factor of 10 was calculated numerically to fit the analytical results with numerical +results. This factor arises from the nonlinear superposition of the fundamental soliton + +components, not considered in Eq. (19) for the sake of simplicity of the analytical formulation. +ECEnew,1 is an analytical expression of ECE that includes the combined effects of interpulse +and intrapulse Raman scattering. The formula of Eq. (24) is therefore a great tool to guide +towards the design of SSFS-based wavelength conversion systems, without requiring heavy +computing approaches. +4. Numerical Validation +The split-step Fourier method (SSFM) is a numerical method which is extensively used for +solving pulse propagation in optical fibers [1]. A recent study by Farag et al. has shown that +the SSFM provides least cumulative error with fastest computational speed compared to other +commonly used numerical methods [24]. For our study, a numerical GNLSE solver is +developed to implement the SSFM and solve the GNLSE in Eq. (2) for pulse evolution, using +MATLAB. To obtain a more accurate simulation output, a full form of the Raman response +function is used instead of using the Raman time constant (TR). The convolution integral in +Eq. (2) is calculated using [25, 26] + +∫ R(T′)|A(z, T − T′)|2dT′ +∞ +0 += (1 − fR)|𝐴(𝑧, 𝑇)|2 + fRΔTℱ−1( ℱ(hR) ℱ(|A(z, T)|2)), +(26) + +where ℱ and ℱ−1 represent the Fourier and inverse Fourier transform operations, respectively, +and ΔT is the temporal grid spacing. Several parameters such as temporal window, temporal +grid spacing, and step size are carefully set in the GNLSE solver to reach a valid solution. +Setting up the proper parameters is essential, especially for modelling MS fission in an optical +fiber, because in a soliton fission process the generated k = 1 Raman soliton is significantly +shorter than the MS, as well as pulses experience large temporal delay and frequency shift, +simultaneously. The temporal window is set such that it contains the maximum temporal delay +of k = 1 Raman soliton over the propagation distance of interest. The temporal grid is sampled +densely enough to ensure that the k = 1 Raman soliton remains defined by several temporal and +spectral points. An adaptive step-size is used to limit the nonlinear phase shift increment in +each propagation step during the simulation. +The GNLSE solver is validated by reproducing temporal and spectral results from literature +[1, 26]. The first simulation consists into the propagation of a second-order soliton in presence +of Raman scattering to simulate SSFS from soliton fission. Fig. 2a shows the generated spectra +which matches closely with Fig. 5.24 in Ref. 1. The second simulation case consists into the +propagation of a femtosecond MS of order ~8.65 before turning into a supercontinuum. In this +simulation, dispersion coefficients up to 10th order and nonlinear parameter up to 1st order are +considered as in Ref. 26. Fig. 2b shows the supercontinuum spectrum which matches closely +with Fig. 4 in Ref. 26. In both cases, the GNLSE solver reproduces the results from the +literature with high accuracy and supports its validity. + + +Fig. 2. Simulation results from the GNLSE solver, comparing with results in literature. (a) Spectral evolution of a +second-order soliton depicting soliton fission and SSFS and (b) supercontinuum generation in the femtosecond regime. + +0.15 +0 +(a) +(b) +Intensity +Distance [m] +-10 +0.1 +Intensity[dB] +0.5 +20 +0 +0.05 +-2 +-30 +0 +8 +6 +2 +2 +4 +0 +-40 +0 +0 +450 +750 +1050 +1350 +Wavelength[nm]Figure 3 shows the ECE results of a k = 1 Raman soliton with different initial pulse +durations and soliton orders obtained from ECEnew, the numerical simulations of Eq. (2) +(ECEGNLSE), and ECEISM. The fiber considered is a lossless SSMF with 2 = - 21.61 ps2/km and +0 = 1.42 W-1/km at the initial pulse wavelength of  = 1.55 m, and 1 = 12.2 fs, 2 = 32 fs, +and fR = 0.18 [18, 27]. +From Fig. 3, it is observed that the ECEISM is independent of pulse duration. This is in +contrast with the ECEnew and ECEGNLSE that diverge gradually from the ISM predictions as the +soliton order increases and pulse duration decreases. The ISM however provides asymptotic +results to the ECEnew and ECEGNLSE as the pulse duration increases. It is also observed that the +ECEnew and ECEGNLSE agree well except for high soliton orders with short pulse durations, +suggesting that another mechanism of energy transfer is triggered. This discrepancy could arise +at a point where the k = 2 soliton would begin to benefit of an enhanced ECE by interpulse +Raman scattering from solitons k > 2. This is expected to arise when the MS order is sufficiently +high and duration is short, such that the spectral overlap between the k = 2 soliton and the +Raman gain function significantly increase. For example, with a MS of N = 6 and +TFWHM = 700 fs, the ECEk = 1 = 53.0% and ECEk = 2 = 22.9%. whereas for a MS with N = 6 and +TFWHM = 400 fs, the ECEk = 1 = 50.0% and ECEk = 2 = 27.2%. The ISM predicts ECE = 25.0% +for the k = 2 soliton. Clearly, the k = 2 soliton eventually benefits of an increased ECE above +the expected ISM as the MS gets shorter. This is however an assumption that requires further +investigation before being validated. + + +Fig. 3. Energy conversion efficiency of k = 1 soliton after fission of high-order solitons as a function of duration and +order. + +Figure 4 shows the influence of 2 and 0 in the ECEGNLSE of a k = 1 Raman soliton +generated from the fission of a MS of N = 3. In the first case, different values of 2 are used to +observe its effect on the ECE. In the second case, different values of 0 are used to see its effects +on the ECE. In both cases, only the peak powers are adjusted to maintain a constant soliton +order. From Figs. 4(a) and 4(b) it is observed that the ECE is independent of 2 and 0 and their +spectral derivative but depend on the MS order and duration. + +ECEnew +ECE(SM +ECE GNLSE +70 +Experiment +N = 2 +·N= 2 +N= 2 +N=3 +·N=3 +N=3 +N=4 +-·N= 4 +N=4 +N= 5 +N= 5 +N= 6 +N=6 +40 +3.0 +500 +1000 +1500 +2000 +2500 +3000 +[fs] + +Fig. 4. ECEGNLSE for the fission generated k = 1 Raman soliton from an MS of N = 3, with variable pulse durations +propagating in fibers. In (a), variable values of GVD coefficient (2) when 0 = 1.42 W-1/km. In (b), variable values of +nonlinear parameter (0) when 2 = - 21.61 ps2/km. + +5. Experimental Validation +Figure 5 shows a schematic of the experimental setup used to probe the ECE of a k = 1 Raman +soliton after soliton fission. A mode-locked fiber laser (Calmar-FPL-02CFFPM) generates seed +pulses centered at 0 = 1.55 m with a duration of 295 fs, an average power of 2.8 mW, and at +the repetition rate of 20 MHz. An in-line polarization controller is used to maximize the SSFS +while an EDFA (Pritel-FA-22-IO) amplifies the seed pulse to an average power of 35.4 mW. +The EDFA also broadens the pulse due to normal dispersion of 35 fs/nm. The amplified pulse +then turns into a MS that experiences soliton fission and SSFS within a 3 m long SSMF patch +cord that leads to a 1% tap coupler (C1). An optical spectrum analyzer (OSA1, Yokogawa- +AQ6376) at the 1% port of the coupler C1 probes the initial SSFS of the k = 1 Raman soliton. +A variable optical attenuator (VOA, Agilent-81577A) after C1 finely controls the energy of +post-fission fundamental soliton before propagation in the nonlinear fiber. The VOA is used as +the variable element of this experiment to monitor the spectrum of the soliton that experiences +the largest amount of SSFS (i.e., the k = 1 Raman soliton). The nonlinear fiber is a 100 m long +SSMF (Thorlabs SMF-28-100), where the soliton experiences SSFS. At the output of the +nonlinear fiber, a coupler (C2) splits the signal into two ports for power and spectrum +monitoring. The power is measured by an optical power meter (Newport-843-R) while the +spectrum is recorded with OSA2 (Yokogawa-AQ6375). For simulation purpose, the +wavelength dependence of the VOA and couplers C1 and C2 are measured in the wavelength +range of 1.5-1.8 m with a broadband source. A constant fiber loss of 0 = 0.2 dB/km within +the wavelength range of interest is considered, as measured using a broadband source. + +Fig. 5. Schematic of the experimental setup. MLFL: mode-locked fiber laser; PC: polarization controller; EDFA: +erbium-doped fiber amplifier; OSA: optical spectrum analyzer; VOA: variable optical attenuator; SSMF: standard +single-mode fiber; PM: power meter. + +At the output of the EDFA, the MS has a pulse duration of 938 fs and N = 5.57 as estimated +from comparison of GNLSE solver and experiment. Figure 6 shows the spectrum measured in +OSA1 of the MS generated from the EDFA. The initial Raman shift of 74 nm within the 3 m +long SSMF between the EDFA and C1 confirms soliton fission and formation of a Raman + +0Z +(a) +70 +(b) +口 +口 +口 +口 +口 +口 +65 +65 +口 +口 +A +A +GNLSE +60 +55 +ECE +ECE +口 +. +50 +50 +45 +FWHM +=900fs +45 ++ +FWHM += 900 fs +2040 +60 +80 +100 +120 +2 +3 +5 +B./[ps/km] +kmMLFLsoliton. Figure 7 shows the obtained spectra as a function of attenuation levels of the VOA, as +well as simulated spectra obtained from numerical evaluation + + + +Fig. 6. Soliton fission and initial SSFS observed in OSA1. + +of Eq. (2). Of particular interest is the spectrum of the k = 1 Raman soliton at long wavelengths +and experiencing a large variation of SSFS as a function of attenuation levels. In the numerical +simulation, GVD coefficients up to 5 and nonlinear coefficient up to  are used as provided +in Table 1. + + + +Fig. 7. (a) Measured output (b) simulated spectra of SSFS with increasing VOA levels. + +Table 1. Taylor coefficients of GVD and nonlinear parameter of SSMF fiber. +GVD +Nonlinear parameter +2: +-21.61 ps2/km +0: +1.42 W-1/km +3: +0.125 ps3/km +1: +0.0029 W-1ps/km +4: +-3.99 × 10-4 ps4/km +2: +-1.43 × 10-6 W-1ps2/km +5: +1.99 × 10-6 ps5/km +3: +-1.11 × 10-8 W-1ps3/km + +The good agreement between numerical simulations and experimental results validates the +GNLSE solver results as well as results presented in Fig. 3. The ECE obtained for the k = 1 +soliton from the experiment is 50.6%, while ECEGNLSE = 51.1% and ECEnew = 53.7%, showing +a good agreement in between those values. In contrast, ECEISM = 32.7%.The results from the +GNLSE solver and the experiments clearly shows an enhanced ECE for the k = 1 Raman soliton +due to interpulse Raman scattering. + + +0 dB +0 dB +-1.4 dB +-1.4 dB +-2.4 dB +-2.4 dB +-3.4 dB +-3.4 dB +-4.4 dB +-4.4 dB6. Effect of Raman Gain Function +According to Eq. (20), the overlap of the spectrum of the k = 1 Raman soliton and the Raman +gain function determines the ECE enhancement of the k = 1 Raman soliton. To observe the +effects of the Raman gain profile on the ECE of k = 1 Raman soliton, numerical simulations +are performed with four different kinds of fiber materials: Silica, ZBLAN, and chalcogenides +(AS2S3 and As2Se3). The Raman functions of silica and chalcogenides are calculated from +Eq. (4). For As2S3, the Raman response function uses 1 = 15.5 fs, 2 = 230.5 fs, and fR = 0.1 +[16]. For As2Se3, 1 = 23.1 fs, 2 = 195 fs [28], and fR = 0.1. An intermediate broadening model +is used to generate the Raman response function of ZBLAN [29]. According to this model, the +Raman response function of ZBLAN is [30] + +hR(T) = ∑ +Ai exp(−γiT) exp (− Γi +2T2 +4 ) sin(ωv,i) T +8 +i=1 +, +(27) + +where the parameters Ai, i, i, and v,i are tabulated in Ref. 30. Figure 8 shows the Raman gain +functions of silica, ZBLAN, and chalcogenides (As2S3 and As2Se3) calculated from their Raman +responses hR(T). The inset of Fig. 8 also presents the Raman time (TR), the maximum Raman +gain coefficient at 1550 nm and the κ parameter for the four materials. + + +Fig. 8. Raman gain functions of Silica, ZBLAN and chalcogenides (As2S3 and As2Se3) normalized with respect to the +peak Raman gain of silica. The inset tabulates the Raman time and maximum Raman gain coefficient at 1550 nm, and +κ parameter of the materials. + +Figure 9 shows the ECEGNLSE and the ECEISM of the k = 1 Raman soliton generated from +the fission of a N = 3 MS in the four fiber materials. From Fig. 9 it is observed that for pulse +durations of the MS larger than 500 fs, the generated k = 1 Raman solitons obtain maximum +amount of ECE in ZBLAN fiber compared to the other three materials. In contrast, the k = 1 +Raman solion gets the minimum amount of ECE in As2S3 fiber. This enhanced ECE in ZBLAN +glass is explained by the maximum Raman gain slope of TR = 4 fs, in contrast with the As2S3 +glass that has the smallest Raman gain slope of TR = 0.2 fs, as shown in Fig. 8. As a result, the +k = 1 Raman soliton experiences more Raman gain in ZBLAN fiber near the pump frequency +than in the other three materials. However, if the pulse duration is short enough (< 1 ps) that the + + +Silica +TR +gR,max +K +- ZBLAN +[fs] +× 10-13 +× 10-13 +lalized + AS,S. +[m/W] +[s] +Silica +3 +0.65 +0.45 +AS,Se. +4 +2 +ZBLAN +4 +0.38 +1.15 +As2S3 +0.2 +3.39 +1.01 +As2Se3 +0.6 +1.91 +1.57 +3 +man +ex +10 +15 +20 +25 +30 +HZ + +Fig. 9. Energy conversion efficiency of the k = 1 Raman soliton after fission of an N = 3 MS of variable pulse durations +and into fibers made of silica, ZBLAN, and chalcogenide glasses. + +broad spectrum of the k = 1 Raman soliton falls within the gain peak of As2S3, the ECE starts +to enhance as can be seen from Fig. 9. From Figs. 8 and 9, it is therefore evident that the impact +of the Raman gain profile is significant on the ECE enhancement of the k = 1 Raman soliton. +Figure 9 also suggests that ZBLAN is the best candidate for obtaining enhanced ECE of the +k = 1 Raman soliton. +7. Conclusion +In summary, we have demonstrated an enhanced energy conversion efficiency of the +k = 1 Raman soliton in a Raman induced soliton fission process. Using the interpulse Raman +scattering model, an analytical formula for the enhanced ECE is derived which provides a good +fit with the numerical simulation results. The GNLSE solver results reveal that the ECE of k = 1 +Raman soliton diverges rapidly from the ISM prediction with increasing soliton order and +decreasing pulse duration. The experimental results agree well with the numerical simulation, +which proves the existence of enhanced ECE for k = 1 Raman soliton from the fission process +triggered by Raman scattering. Finally, ECE results of the k = 1 Raman soliton in four different +kinds of fibers show the impact of the Raman gain profile on the ECE enhancement and suggest +that ZBLAN is the best material for wavelength conversion using soliton fission. This study is +useful for understanding the physical process behind the enhancement of ECE of the k = 1 +Raman soliton in a soliton fission process. Moreover, it is useful for the design of optical +devices making use of soliton fission, such as SSFS based wavelength converters and +supercontinuum sources. +Data Availability. Data underlying the results presented in this paper are not publicly +available at this time but may be obtained from the authors upon reasonable request. +Funding. Natural Sciences and Engineering Research Council of Canada. +Disclosures. The authors declare no conflicts of interest. +References +1. +G. P. Agrawal, Nonlinear Fiber Optics, 6th ed. (Elsevier, 2019). +2. +R. Kormokar, M. Shamim, and M. 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B 29, 238–243 (2012). + diff --git a/V9E1T4oBgHgl3EQfJAMR/content/tmp_files/load_file.txt b/V9E1T4oBgHgl3EQfJAMR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..94f11a33ee41bbc1dcaeb6c2d50fc9e1e90f1e56 --- /dev/null +++ b/V9E1T4oBgHgl3EQfJAMR/content/tmp_files/load_file.txt @@ -0,0 +1,644 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf,len=643 +page_content='Energy conversion efficiency from a high order soliton to fundamental solitons in presence of Raman scattering ROBI KORMOKAR,* MD HOSNE MOBAROK SHAMIM, AND MARTIN ROCHETTE Department of Electrical and Computer Engineering, McGill University, 3480 University Street, Montréal, Québec, H3A 0E9, Canada *robi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='kormokar@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='mcgill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='ca Abstract: We formulate the energy conversion efficiency from a high-order soliton to fundamental solitons by including the influence of interpulse Raman scattering in the fission process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The proposed analytical formula agrees closely with numerical results of the generalized nonlinear Schrödinger equation as well as to experimental results, while the resulting formulation significantly alters the energy conversion efficiency predicted by the Raman-independent inverse scattering method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' We also calculate the energy conversion efficiency in materials of different Raman gain profiles such as silica, ZBLAN and chalcogenide glasses (As2S3 and As2Se3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' It is predicted that ZBLAN glass leads to the largest energy conversion efficiency of all four materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The energy conversion efficiency is a notion of utmost practical interest for the design of wavelength converters and supercontinuum generation systems based on the dynamics of soliton self-frequency shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Introduction Soliton self-frequency shift (SSFS) is an intrapulse Raman scattering process that is key for wavelength conversion and supercontinuum generation [1-7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Among the several nonlinear wavelength conversion mechanisms, SSFS is of particular interest because of its wide tunability range beyond tens of THz [8-11], and the absence of phase-matching constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' SSFS is best observed with short pulses, as SSFS increases proportionally with the fourth power of inverse pulse duration [2, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' This is particularly the case when an Nth order soliton or mother soliton (MS) experiences fission and splits into N fundamental solitons, from which solitons of femtosecond duration typically emerge [1, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Exploiting the soliton fission mechanism for SSFS based wavelength conversion has been reported by different groups [14-16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' For example, Gauthier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' reported the soliton fission of a MS into solitons experiencing SSFS for wavelength conversion up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='8 \uf06dm in InF3 fiber [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Alamgir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' reported continuously tunable Raman solitons over the frequency range of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='047-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='667 \uf06dm using soliton fission and SSFS in an As2S3 microwire [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Soliton fission is also a fundamental mechanism for supercontinuum generation in fiber with anomalous dispersion [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The energy conversion efficiency (ECE) from the MS to each fundamental soliton is of practical interest for wavelength conversion applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' For instance, one may wish to optimize the energy transfer from the MS specifically towards the fundamental soliton that experiences the largest amount of SSFS [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Using the inverse scattering method (ISM), Kodama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' have shown that an Nth order soliton is a bound state of N fundamental solitons, as a solution to the nonlinear Schrödinger equation (NLSE) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' They have also shown analytically that the bound state Nth order soliton is breaking, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=', experiences fission, under sufficient high-order linear and/or nonlinear perturbation in the fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' According to the ISM, each fundamental soliton after fission carries an energy given by [1, 13, 17] Ek = (2N + 1 − 2k) N2 E0, (1) where N = (\uf0670E0T0/2|\uf0622|)1/2 with E0 and T0 being the energy and duration of the MS, respectively, \uf0622 is the group velocity dispersion (GVD) coefficient, and \uf0670 is the waveguide nonlinear parameter at center frequency \uf0770.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The index k = 1…N labels each soliton emerging from fission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Each soliton thus carries an energy Ek given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 1, leading to an ECE given by ECEISM,k = (100Ek/E0)%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Of particular interest for wavelength conversion is the k = 1 Raman soliton because it is the shortest, the most powerful, and most importantly, it experiences the largest amount of SSFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' We however recently noticed numerically and experimentally that the k = 1 Raman soliton emerging from soliton fission often carries significantly more energy than predicted by the ISM, leading to ECE > ECEISM,1 [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' In this paper, we demonstrate that the ECE evaluated from numerical simulations and from experiment agree together but diverge from the theoretical prediction of ECEISM,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' We subsequently propose an improved analytical formulation of ECE for the k = 1 Raman soliton, derived by adding an energy transfer contribution from interpulse Raman scattering in between the k = 2…N solitons to the k = 1 soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' We show that the ECEISM,1 diverges rapidly from numerical predictions and experiment with MS of increasing order and decreasing duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' This study clarifies the dynamics of Raman induced soliton fission, as well as providing a more precise evaluation of ECE, a topic of utmost interest for the design of SSFS based wavelength converters and supercontinuum sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Theory of Enhanced Energy Conversion Efficiency 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='1 Generalized Nonlinear Schrödinger Equation The propagation of an optical pulse in a nonlinear and dispersive optical fiber is well described by the generalized nonlinear Schrödinger equation (GNLSE) [1] ∂A ∂z + 1 2 (α0 + ∑ inαn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' ∂n ∂Tn ∞ n=1 ) A − i ∑ inβn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' ∂nA ∂Tn ∞ n=2 = i (γ0 + ∑ inγn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' ∂n ∂Tn ∞ n=1 ) (A(z, T) ∫ R(T′)|A(z, T − T′)|2dT′ ∞ 0 ), (2) where A is the slowly varying pulse envelope, \uf061n = dn\uf061/d\uf077n, \uf062n = dn\uf062/d\uf077n, \uf067n = dn\uf067/d\uf077n are the Taylor coefficients of the frequency dependent fiber loss (\uf061), propagation constant (\uf062) and nonlinear parameter (\uf067), respectively, evaluated at the center frequency \uf0770.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' T is time in a frame of reference moving at group velocity of the pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' R(T) is the nonlinear response function defined as [1] R(T) = (1 − fR)δ(T) + fRhR(T), (3) where fR represents the fractional contribution of the delayed Raman response to nonlinear polarization, \uf064(T) is the Dirac delta function, and hR(T) is the Raman response function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Using the single peak Lorentzian model, hR(T) can be written as hR(T) = (τ1 −2 + τ2 −2)τ1 exp(− T τ2 ⁄ ) sin(T τ1 ⁄ ), (4) where \uf057R = 1/\uf0741 is the angular frequency of the Raman gain peak and \uf047R = 1/\uf0742 is the bandwidth of the Raman gain response [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='2 Soliton Constitution and Fission By ignoring high-order linear and nonlinear terms and by considering a lossless medium, the NLSE can be written in a normalized form ∂q ∂Z = i 2 ∂2q ∂τ2 + i|q|2q, (5) with q = N √P0 A, Z = z LD , and τ = T T0 , (6) where P0 and T0 are the pulse peak power and duration, and LD = T0 2/|\uf0622| is the dispersion length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' By using the inverse scattering method developed by Zakharov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' [19], Satsuma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' showed that a solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (5) is an Nth order soliton of the form q = Nsech(\uf074), which itself contains N fundamental solitons with amplitude [20] ηk = 2(N − k) + 1, k = 1, 2, … N (7) and where fundamental solitons are expressed as [17] qk(τ, Z) = ηk sech[ηk(τ + κkZ − τ0k)] exp [−iκkτ + i 2 (ηk 2 − κk 2)Z − iσ0k], (8) where \uf06bk is related with the change in group velocity of the kth soliton component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Satsuma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' showed that \uf06bk = 0 and all fundamental solitons of a high-order soliton travel at the same group velocity, leading to a bound state of solitons [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Because of the phase interference among fundamental solitons, the resulting high-order soliton profile and spectrum evolve periodically with a period of Z = \uf070/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' By using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (6) and (7) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (8) and by setting the constant phase term \uf0730k and the temporal offset of each soliton \uf0740k to zero, the fundamental solitons are expressed as Ak(T, 0) = (2N + 1 − 2k) N √P0 sech [ T T0 (2N + 1 − 2k) ⁄ ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (9) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (9), the peak power and duration of each fundamental soliton component is Pk = (2N + 1 − 2k)2 N2 P0, (10a) and Tk = T0 (2N + 1 − 2k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (10b) A high-order soliton becomes however unstable and eventually fissions under the influence of linear and/or nonlinear perturbations such as Raman effect, third-order dispersion, and self- steepening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' As a result, fundamental solitons come apart and propagate with a peak power and pulse duration defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' According to the ISM, the ECE of each soliton is given by ECEISM,k = (2N + 1 − 2k) N2 × 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (11) In a glassy medium such as an optical fiber, soliton propagation is also accompanied with intrapulse Raman scattering that gradually transfers energy from short wavelength contents of the pulse to the long wavelengths [2, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' As a result, the soliton experiences SSFS following [2] Ω(z) = − 8TR|β2| 15T0 4 zeff, (12) where Ω is SSFS in rad/s, TR is Raman time, and zeff = 1 4α (1 − e−4αz), (13) is the effective length of the fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Equation (12) shows that SSFS increases proportionally with the effective fiber length and with inverse pulse duration to the fourth order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Since the k = 1 soliton has the shortest duration, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (10b), this soliton therefore experiences the largest amount of SSFS compared to the other soliton components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Figure 1 shows a typical example of Raman induced soliton fission process of a N = 3 MS in a 15 m long lossless standard single mode fiber (SSMF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' To demonstrate the concept of Raman induced soliton fission, the simulation only considers the Raman effect as the sole high-order nonlinear effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Fiber parameters are \uf0622 = - 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='61 ps2/km, and \uf0670 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='42 W-1/km at the initial pulse wavelength of \uf06c0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='55 \uf06dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The Raman response function given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (4) is used for silica with \uf0741 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='2 fs, \uf0742 = 32 fs and fR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The initial pulse duration and peak power are 283 fs and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='7 kW, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The simulation is performed by numerically solving the GNLSE of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (2) using a split-step Fourier method (details in section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 1, the k = 1 Raman soliton emerges due to Raman induced soliton fission and experiences the most SSFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' It is also observed that ECEk=1 = 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='0% whereas ECEISM,1 = 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The numerical simulations and ISM results thus significantly differ in this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Simulated (a) temporal and (b) spectral evolution of a third-order MS following a Raman induced soliton fission process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The pulse initially has a center wavelength of \uf06c0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='55 \uf06dm, pulse duration of 283 fs and peak power of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='7 kW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The medium is a 15 m long lossless SSMF with \uf0622 = - 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='61 ps2/km, \uf0670 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='42 W-1/km at \uf06c0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='55 \uf06dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The observed ECE enhancement with respect to the ISM prediction is explained from a transfer of energy in between solitons following the fission process, known as interpulse Raman scattering [21, 22], and in contrast with intrapulse Raman scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' In a bound state, a high- order soliton experiences a periodic evolution due to a mutual interaction between GVD and self-phase modulation (SPM) [17, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' High-order linear and/or nonlinear perturbations break 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='2 (a) (b) Input Input 4 Output Output 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='8 k = 1 soliton 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='6 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='4 k = 1 soliton 人 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='2 01 5 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='7down the bound state, which triggers a fission process, and the MS breaks into fundamental soliton components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' In any glassy medium, Raman scattering is the dominant mechanism that triggers soliton fission [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The Raman effect transfers soliton energy towards lower frequency components via intra- and interpulse Raman scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Intrapulse Raman scattering is responsible for SSFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Interpulse Raman scattering also occurs after the fission event, amplifying solitons at long wavelength from solitons at short wavelengths, leading to energy transfer in between solitons that temporally overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Since the k = 1 Raman soliton experiences the most redshift from SSFS, it becomes amplified by the remaining k = 2…N solitons for as long as they temporally overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' For this reason, the k = 1 Raman soliton is expected to get an enhanced ECE with respect to the ISM prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Note that for a MS with N being a non- integer value, soliton fission is followed by fundamental solitons and dispersive waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Analytical Formulation of Enhanced ECE In addition to the ISM prediction of ECE that occurs during the fission of the MS, a new formulation of ECE should also consider the energy transfer (ΔE) from the k = 2…N solitons to the k = 1 Raman soliton resulting from interpulse Raman scattering that occurs moments after fission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Taking interpulse Raman scattering into account, the new ECE of the k = 1 soliton is expressed as ECEnew,1 = ECEISM,1 + Δe, (14) where Δe is a correction term to the ECEISM,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' For interpulse Raman scattering to occur, the temporal overlap in between involved pulses is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' An estimated pump energy from each k = 2…N solitons to the k = 1 Raman soliton is expressed as Epk(z) = ∫ Pk sech2 ( T Tk ) dT 3T1+q −3T1+q , (15) where T1 and q are the pulse duration and temporal position of the k = 1 Raman soliton, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Pk and Tk are the peak power and pulse duration of the k = 2…N solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The integration limits in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (15) are set such that when q = 0 the integration provides the energies of the k = 2…N solitons that are overlapped with 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='0% of the total energy of the k = 1 soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (10) and integration, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (15) transforms into Epk(z) = E0(2N + 1 − 2k) N2 tanh [3(2N + 1 − 2k) (2N − 1) ] Λk(q), (16a) where Λk(q) = (1 − tanh2 q Tk) (1 − tanh2 3T1 Tk tanh2 q Tk) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (16b) The temporal position of the k = 1 soliton is evaluated from the moment method [23] q(z) = 4TR|β2|2 15T1 4 z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (17) Therefore, the total pump energy from the k = 2…N soliton components sums up into EpT(z) = ∑ E0(2N + 1 − 2k) N2 tanh [3(2N + 1 − 2k) (2N − 1) ] Λk(q) N k=2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (18) The ECE enhancement rate with propagation distance is proportional to EpT (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Therefore, de 𝑑𝑧 ∝ ∑ ( (2N + 1 − 2k) N2 tanh [3(2N + 1 − 2k) (2N − 1) ]) Λk(q) N k=2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (19) This rate also depends on the effective Raman gain experienced by the k = 1 Raman soliton, quantified as de 𝑑𝑧 ∝ γ0P1fR ∫ Im[h̃R(ω)] sech2 [π(ω − Ω(z))T0 2(2N − 1) ] dω 0 −∞ , (20) where P1 is the peak power of the k = 1 Raman soliton, and Im[h̃R(ω)] is the Raman gain function [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' \uf057(z) is the SSFS experienced by the k = 1 Raman soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' By combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (19) and (20), the ECE enhancement rate becomes de 𝑑𝑧 ∝ ∑ ( (2N + 1 − 2k) N2 tanh [3(2N + 1 − 2k) (2N − 1) ]) Λk(q) N k=2 × γ0P1fR ∫ Im[h̃R(ω)] sech2 [π(ω − Ω(z))T0 2(2N − 1) ] dω 0 −∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (21) By integration of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (21) with respect to z, the correction term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (14) becomes Δe ∝ ∑ Ik ( (2N + 1 − 2k) N2 tanh [3(2N + 1 − 2k) (2N − 1) ]) N k=2 , (22) where Ik(z) = ∫ Λk(q)γ0P1fR ∫ [Im[h̃R(ω)] sech2 [π(ω − Ω(z))T0 2(2N − 1) ]] dω 0 −∞ dz z 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (23) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (14) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (22), the correction to the ECE predicted by the ISM becomes ECEnew,1 = ECEISM,1 [1 + κ ∑ Ik (2N + 1 − 2k) (2N − 1) tanh (3(2N + 1 − 2k) 2N − 1 ) N k=2 ], (24) including a material dependent proportionality constant κ = 10 ∫ Im[h̃R(ω)]dω 0 −∞ (25) where the factor of 10 was calculated numerically to fit the analytical results with numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' This factor arises from the nonlinear superposition of the fundamental soliton components, not considered in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (19) for the sake of simplicity of the analytical formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' ECEnew,1 is an analytical expression of ECE that includes the combined effects of interpulse and intrapulse Raman scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The formula of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (24) is therefore a great tool to guide towards the design of SSFS-based wavelength conversion systems, without requiring heavy computing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Numerical Validation The split-step Fourier method (SSFM) is a numerical method which is extensively used for solving pulse propagation in optical fibers [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' A recent study by Farag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' has shown that the SSFM provides least cumulative error with fastest computational speed compared to other commonly used numerical methods [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' For our study, a numerical GNLSE solver is developed to implement the SSFM and solve the GNLSE in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (2) for pulse evolution, using MATLAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' To obtain a more accurate simulation output, a full form of the Raman response function is used instead of using the Raman time constant (TR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The convolution integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (2) is calculated using [25, 26] ∫ R(T′)|A(z, T − T′)|2dT′ ∞ 0 = (1 − fR)|𝐴(𝑧, 𝑇)|2 + fRΔTℱ−1( ℱ(hR) ℱ(|A(z, T)|2)), (26) where ℱ and ℱ−1 represent the Fourier and inverse Fourier transform operations, respectively, and ΔT is the temporal grid spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Several parameters such as temporal window, temporal grid spacing, and step size are carefully set in the GNLSE solver to reach a valid solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Setting up the proper parameters is essential, especially for modelling MS fission in an optical fiber, because in a soliton fission process the generated k = 1 Raman soliton is significantly shorter than the MS, as well as pulses experience large temporal delay and frequency shift, simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The temporal window is set such that it contains the maximum temporal delay of k = 1 Raman soliton over the propagation distance of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The temporal grid is sampled densely enough to ensure that the k = 1 Raman soliton remains defined by several temporal and spectral points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' An adaptive step-size is used to limit the nonlinear phase shift increment in each propagation step during the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The GNLSE solver is validated by reproducing temporal and spectral results from literature [1, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The first simulation consists into the propagation of a second-order soliton in presence of Raman scattering to simulate SSFS from soliton fission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 2a shows the generated spectra which matches closely with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='24 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The second simulation case consists into the propagation of a femtosecond MS of order ~8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='65 before turning into a supercontinuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' In this simulation, dispersion coefficients up to 10th order and nonlinear parameter up to 1st order are considered as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 2b shows the supercontinuum spectrum which matches closely with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 4 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' In both cases, the GNLSE solver reproduces the results from the literature with high accuracy and supports its validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Simulation results from the GNLSE solver, comparing with results in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (a) Spectral evolution of a second-order soliton depicting soliton fission and SSFS and (b) supercontinuum generation in the femtosecond regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='15 0 (a) (b) Intensity Distance [m] -10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='1 Intensity[dB] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='5 20 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='05 -2 -30 0 8 6 2 2 4 0 -40 0 0 450 750 1050 1350 Wavelength[nm]Figure 3 shows the ECE results of a k = 1 Raman soliton with different initial pulse durations and soliton orders obtained from ECEnew, the numerical simulations of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (2) (ECEGNLSE), and ECEISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The fiber considered is a lossless SSMF with \uf0622 = - 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='61 ps2/km and \uf0670 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='42 W-1/km at the initial pulse wavelength of \uf06c\uf030 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='55 \uf06dm, and \uf0741 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='2 fs, \uf0742 = 32 fs, and fR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='18 [18, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 3, it is observed that the ECEISM is independent of pulse duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' This is in contrast with the ECEnew and ECEGNLSE that diverge gradually from the ISM predictions as the soliton order increases and pulse duration decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The ISM however provides asymptotic results to the ECEnew and ECEGNLSE as the pulse duration increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' It is also observed that the ECEnew and ECEGNLSE agree well except for high soliton orders with short pulse durations, suggesting that another mechanism of energy transfer is triggered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' This discrepancy could arise at a point where the k = 2 soliton would begin to benefit of an enhanced ECE by interpulse Raman scattering from solitons k > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' This is expected to arise when the MS order is sufficiently high and duration is short, such that the spectral overlap between the k = 2 soliton and the Raman gain function significantly increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' For example, with a MS of N = 6 and TFWHM = 700 fs, the ECEk = 1 = 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='0% and ECEk = 2 = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='9%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' whereas for a MS with N = 6 and TFWHM = 400 fs, the ECEk = 1 = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='0% and ECEk = 2 = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The ISM predicts ECE = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='0% for the k = 2 soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Clearly, the k = 2 soliton eventually benefits of an increased ECE above the expected ISM as the MS gets shorter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' This is however an assumption that requires further investigation before being validated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Energy conversion efficiency of k = 1 soliton after fission of high-order solitons as a function of duration and order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Figure 4 shows the influence of \uf0622 and \uf0670 in the ECEGNLSE of a k = 1 Raman soliton generated from the fission of a MS of N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' In the first case, different values of \uf0622 are used to observe its effect on the ECE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' In the second case, different values of \uf0670 are used to see its effects on the ECE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' In both cases, only the peak powers are adjusted to maintain a constant soliton order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' From Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 4(a) and 4(b) it is observed that the ECE is independent of \uf0622 and \uf0670 and their spectral derivative but depend on the MS order and duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' ECEnew ECE(SM ECE GNLSE 70 Experiment N = 2 N= 2 N= 2 N=3 N=3 N=3 N=4 N= 4 N=4 N= 5 N= 5 N= 6 N=6 40 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='0 500 1000 1500 2000 2500 3000 [fs] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' ECEGNLSE for the fission generated k = 1 Raman soliton from an MS of N = 3, with variable pulse durations propagating in fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' In (a), variable values of GVD coefficient (\uf0622) when \uf0670 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='42 W-1/km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' In (b), variable values of nonlinear parameter (\uf0670) when \uf0622 = - 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='61 ps2/km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Experimental Validation Figure 5 shows a schematic of the experimental setup used to probe the ECE of a k = 1 Raman soliton after soliton fission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' A mode-locked fiber laser (Calmar-FPL-02CFFPM) generates seed pulses centered at \uf06c0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='55 \uf06dm with a duration of 295 fs, an average power of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='8 mW, and at the repetition rate of 20 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' An in-line polarization controller is used to maximize the SSFS while an EDFA (Pritel-FA-22-IO) amplifies the seed pulse to an average power of 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='4 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The EDFA also broadens the pulse due to normal dispersion of 35 fs/nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The amplified pulse then turns into a MS that experiences soliton fission and SSFS within a 3 m long SSMF patch cord that leads to a 1% tap coupler (C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' An optical spectrum analyzer (OSA1, Yokogawa- AQ6376) at the 1% port of the coupler C1 probes the initial SSFS of the k = 1 Raman soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' A variable optical attenuator (VOA, Agilent-81577A) after C1 finely controls the energy of post-fission fundamental soliton before propagation in the nonlinear fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The VOA is used as the variable element of this experiment to monitor the spectrum of the soliton that experiences the largest amount of SSFS (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=', the k = 1 Raman soliton).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The nonlinear fiber is a 100 m long SSMF (Thorlabs SMF-28-100), where the soliton experiences SSFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' At the output of the nonlinear fiber, a coupler (C2) splits the signal into two ports for power and spectrum monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The power is measured by an optical power meter (Newport-843-R) while the spectrum is recorded with OSA2 (Yokogawa-AQ6375).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' For simulation purpose, the wavelength dependence of the VOA and couplers C1 and C2 are measured in the wavelength range of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='5-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='8 \uf06dm with a broadband source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' A constant fiber loss of \uf0610 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='2 dB/km within the wavelength range of interest is considered, as measured using a broadband source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Schematic of the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' MLFL: mode-locked fiber laser;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' PC: polarization controller;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' EDFA: erbium-doped fiber amplifier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' OSA: optical spectrum analyzer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' VOA: variable optical attenuator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' SSMF: standard single-mode fiber;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' PM: power meter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' At the output of the EDFA, the MS has a pulse duration of 938 fs and N = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='57 as estimated from comparison of GNLSE solver and experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Figure 6 shows the spectrum measured in OSA1 of the MS generated from the EDFA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The initial Raman shift of 74 nm within the 3 m long SSMF between the EDFA and C1 confirms soliton fission and formation of a Raman 0Z (a) 70 (b) 口 口 口 口 口 口 65 65 口 口 A A GNLSE 60 55 ECE ECE 口 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 50 50 45 FWHM =900fs 45 + FWHM = 900 fs 2040 60 80 100 120 2 3 5 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='/[ps/km] kmMLFLsoliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Figure 7 shows the obtained spectra as a function of attenuation levels of the VOA, as well as simulated spectra obtained from numerical evaluation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Soliton fission and initial SSFS observed in OSA1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Of particular interest is the spectrum of the k = 1 Raman soliton at long wavelengths and experiencing a large variation of SSFS as a function of attenuation levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' In the numerical simulation, GVD coefficients up to \uf0625 and nonlinear coefficient up to \uf067\uf033 are used as provided in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (a) Measured output (b) simulated spectra of SSFS with increasing VOA levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Taylor coefficients of GVD and nonlinear parameter of SSMF fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' GVD Nonlinear parameter \uf0622: -21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='61 ps2/km \uf0670: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='42 W-1/km \uf0623: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='125 ps3/km \uf0671: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='0029 W-1ps/km \uf0624: -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='99 × 10-4 ps4/km \uf0672: -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='43 × 10-6 W-1ps2/km \uf0625: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='99 × 10-6 ps5/km \uf0673: -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='11 × 10-8 W-1ps3/km The good agreement between numerical simulations and experimental results validates the GNLSE solver results as well as results presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The ECE obtained for the k = 1 soliton from the experiment is 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='6%, while ECEGNLSE = 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='1% and ECEnew = 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='7%, showing a good agreement in between those values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' In contrast, ECEISM = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='The results from the GNLSE solver and the experiments clearly shows an enhanced ECE for the k = 1 Raman soliton due to interpulse Raman scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 0 dB 0 dB -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='4 dB -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='4 dB -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='4 dB -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='4 dB -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='4 dB -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='4 dB -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='4 dB -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='4 dB6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Effect of Raman Gain Function According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (20), the overlap of the spectrum of the k = 1 Raman soliton and the Raman gain function determines the ECE enhancement of the k = 1 Raman soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' To observe the effects of the Raman gain profile on the ECE of k = 1 Raman soliton, numerical simulations are performed with four different kinds of fiber materials: Silica, ZBLAN, and chalcogenides (AS2S3 and As2Se3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The Raman functions of silica and chalcogenides are calculated from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' For As2S3, the Raman response function uses \uf0741 = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='5 fs, \uf0742 = 230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='5 fs, and fR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='1 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' For As2Se3, \uf0741 = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='1 fs, \uf0742 = 195 fs [28], and fR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' An intermediate broadening model is used to generate the Raman response function of ZBLAN [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' According to this model, the Raman response function of ZBLAN is [30] hR(T) = ∑ Ai exp(−γiT) exp (− Γi 2T2 4 ) sin(ωv,i) T 8 i=1 , (27) where the parameters Ai, \uf067i, \uf047i, and \uf077v,i are tabulated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Figure 8 shows the Raman gain functions of silica, ZBLAN, and chalcogenides (As2S3 and As2Se3) calculated from their Raman responses hR(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 8 also presents the Raman time (TR), the maximum Raman gain coefficient at 1550 nm and the κ parameter for the four materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Raman gain functions of Silica, ZBLAN and chalcogenides (As2S3 and As2Se3) normalized with respect to the peak Raman gain of silica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The inset tabulates the Raman time and maximum Raman gain coefficient at 1550 nm, and κ parameter of the materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Figure 9 shows the ECEGNLSE and the ECEISM of the k = 1 Raman soliton generated from the fission of a N = 3 MS in the four fiber materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 9 it is observed that for pulse durations of the MS larger than 500 fs, the generated k = 1 Raman solitons obtain maximum amount of ECE in ZBLAN fiber compared to the other three materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' In contrast, the k = 1 Raman solion gets the minimum amount of ECE in As2S3 fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' This enhanced ECE in ZBLAN glass is explained by the maximum Raman gain slope of TR = 4 fs, in contrast with the As2S3 glass that has the smallest Raman gain slope of TR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='2 fs, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' As a result, the k = 1 Raman soliton experiences more Raman gain in ZBLAN fiber near the pump frequency than in the other three materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' However, if the pulse duration is short enough (< 1 ps) that the Silica TR gR,max K ZBLAN [fs] × 10 13 × 10 13 lalized AS,S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' [m/W] [s] Silica 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='45 AS,Se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 4 2 ZBLAN 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='38 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='15 As2S3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='39 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='01 As2Se3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='91 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content='57 3 man ex 10 15 20 25 30 HZ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Energy conversion efficiency of the k = 1 Raman soliton after fission of an N = 3 MS of variable pulse durations and into fibers made of silica, ZBLAN, and chalcogenide glasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' broad spectrum of the k = 1 Raman soliton falls within the gain peak of As2S3, the ECE starts to enhance as can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' From Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 8 and 9, it is therefore evident that the impact of the Raman gain profile is significant on the ECE enhancement of the k = 1 Raman soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Figure 9 also suggests that ZBLAN is the best candidate for obtaining enhanced ECE of the k = 1 Raman soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Conclusion In summary, we have demonstrated an enhanced energy conversion efficiency of the k = 1 Raman soliton in a Raman induced soliton fission process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Using the interpulse Raman scattering model, an analytical formula for the enhanced ECE is derived which provides a good fit with the numerical simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The GNLSE solver results reveal that the ECE of k = 1 Raman soliton diverges rapidly from the ISM prediction with increasing soliton order and decreasing pulse duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The experimental results agree well with the numerical simulation, which proves the existence of enhanced ECE for k = 1 Raman soliton from the fission process triggered by Raman scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Finally, ECE results of the k = 1 Raman soliton in four different kinds of fibers show the impact of the Raman gain profile on the ECE enhancement and suggest that ZBLAN is the best material for wavelength conversion using soliton fission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' This study is useful for understanding the physical process behind the enhancement of ECE of the k = 1 Raman soliton in a soliton fission process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Moreover, it is useful for the design of optical devices making use of soliton fission, such as SSFS based wavelength converters and supercontinuum sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Data Availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Natural Sciences and Engineering Research Council of Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Disclosures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' The authors declare no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' Agrawal, Nonlinear Fiber Optics, 6th ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E1T4oBgHgl3EQfJAMR/content/2301.02945v1.pdf'} +page_content=' (Elsevier, 2019).' metadata={'source': 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For a triangulated category T , Matsui recently introduced a topo- +logical space Spec△(T ) which we call the triangular spectrum of T as an analog +of the Balmer spectrum introduced by Balmer for a tensor triangulated category. +In the present paper, we use the triangular spectrum to reconstruct a noetherian +scheme X from its perfect derived category Dpf(X). As an application, we give an +alternative proof of the Bondal-Orlov-Ballard reconstruction theorem. Moreover, +we define the structure sheaf on Spec△(T ) and compare the triangular spectrum +and the Balmer spectrum as ringed spaces. +1. Introduction +In this paper, we consider the reconstruction problem of noetherian schemes from +their perfect derived categories: does a triangle equivalence between perfect derived +categories Dpf(X) ∼= Dpf(Y ) imply an isomorphism X ∼= Y of noetherian schemes? +Many authors have studied this kind of reconstruction problem well; see [1, 2, 9, 10, +20, 21]. It is well-known that affine noetherian schemes are reconstructed using the +triangulated category structures from their perfect derived categories. By contrast, +this reconstruction problem fails for non-affine noetherian schemes in general. For +example, Mukai [18] proved that for an abelian variety A, A and its dual A∨ (which +are not isomorphic in general) have the equivalent perfect derived categories. +Therefore, the triangulated category structure is insufficient for reconstructing X +from Dpf(X). Balmer proved that X could be reconstructed from Dpf(X) using +the tensor triangulated category structure as follows. For an essentially small tensor +triangulated category (T , ⊗, 1), Balmer defined the ringed space Spec⊗(T ), which we +call the Balmer spectrum of (T , ⊗, 1). In [3], he proved that there is an isomorphism +X ∼= Spec⊗(Dpf(X)) +for the tensor triangulated category (Dpf(X), ⊗L +OX, OX). This isomorphism shows +that X is reconstructed from Dpf(X) using the tensor triangulated category struc- +ture. Balmer spectra allow us to study tensor triangulated categories via algebro- +geometric methods. This theory is called tensor triangular geometry and has been +actively studied in various fields of mathematics. Recently, Matsui [17] introduced +the topological space Spec△(T ) for a triangulated category T to generalize tensor +2020 Mathematics Subject Classification. 14A15, 14F08, 14H52, 18G80. +Key words and phrases. Balmer spectrum, elliptic curve, noetherian scheme, perfect derived +category, tensor triangulated category, triangular spectrum, triangulated category. +The author was partly supported by JSPS Grant-in-Aid for Early-Career Scientists 22K13894. +1 + +2 +HIROKI MATSUI +triangular geometry to triangulated categories. We call Spec△(T ) the triangular +spectrum of T . For the perfect derived category Dpf(X) of a noetherian scheme X, +it is shown in [17] that there is an immersion +X ֒→ Spec△(Dpf(X)) +of topological spaces. +One of the most famous reconstruction results is due to Bondal-Orlov [9] and +Ballard [1], which states that for Gorenstein projective varieties X1 and X2 over a +field k with ample or anti-ample canonical bundles, X1 and X2 are isomorphic as +varieties whenever their perfect derived categories are equivalent as k-linear trian- +gulated categories. As an application of triangular spectra, we prove the following +result, which generalizes the Bondal-Orlov-Ballard reconstruction theorem. +Theorem 1.1. (Theorem 3.3). Let X1 and X2 be noetherian schemes, and Φ : +Dpf(X1) +∼ +=−→ Dpf(X2) be a triangle equivalence. Assume there is a line bundle Li on +Xi for i = 1, 2 satisfying the following conditions: +(i) Li is ample or anti-ample for i = 1, 2. +(ii) There is an isomorphism +Φ(F ⊗L +OX1 L1) ∼= Φ(F) ⊗L +OX2 L2 +for any F ∈ Dpf(X1). +Then (X1)red and (X2)red are isomorphic as schemes. +Actually, the reconstruction of underlying topological spaces essentially appeared in +[12]. Although they use the result of Koll´ar-Lieblich-Olsson-Sawin [15] to reconstruct +structure sheaves, in this paper, we give a more elementary and algebraic proof for +it. To reconstruct the structure sheaf, the center of a triangulated category plays +an important role. +So far, we consider the triangular spectrum Spec△(T ) as a topological space. This +paper introduces the structure sheaf on Spec△(T ) for a triangulated category T and +makes Spec△(T ) the ringed space. The following second main theorem compares +the Balmer spectrum and the triangular spectrum as ringed spaces: +Theorem 1.2. (Theorem 4.6). Let (T , ⊗, 1) be an idempotent complete, rigid, and +locally monogenic tensor triangulated category. Assume further that Spec⊗(T ) is +noetherian. Then there is a morphism +i : Spec⊗(T )red → Spec△(T ) +of ringed spaces satisfying the following conditions: +(1) The map of underlying topological spaces is the inclusion. +(2) i is an open immersion of ringed spaces whenever Spec⊗(T ) is an open subset +of Spec△(T ). +(3) i is an isomorphism of ringed spaces if T is generated by 1. +Here, a tensor triangulated category (T , ⊗, 1) is said to be rigid if it is closed and +every object is strongly dualizable. (T , ⊗, 1) is said to be is locally monogenic if it +is locally generated by the unit object; see Section 3 for details. + +TRIANGULAR SPECTRA AND DERIVED CATEGORIES +3 +The assumptions in Theorem 1.2 are satisfied for the perfect derived categories of +noetherian schemes. Applying this theorem to such tensor triangulated categories, +we obtain the following result: +Theorem 1.3. (Corollaries 4.7, 4.9, and 4.10). +(1) Let X be a noetherian quasi-affine scheme. Then there is an isomorphism +Spec△(Dpf(X)) ∼= Xred +of ringed spaces. +(2) Let P1 be the projective line over a field k. Then there is an isomorphism +Spec△(Dpf(P1)) ∼= P1 ⊔ +�� +n∈Z +Spec(k) +� +of ringed spaces. +(3) Let E be an elliptic curve over an algebraically closed field. Then there is an +isomorphism +Spec△(Dpf(E)) ∼= E ⊔ + + � +(r,d)∈I +Er,d + + +of ringed spaces. Here, I := {(r, d) ∈ Z2 | r > 0, gcd(r, d) = 1}, and Er,d is a +copy of E for each (r, d) ∈ I. +Matsui [17](1),(2) and Hirano-Ouchi [12](3) proved that there are homeomorphisms +between the underlying spaces. +Therefore, Theorem 1.3 extends their result to +isomorphisms of ringed spaces. Theorem 1.3(1)(3) shows that reduced quasi-affine +schemes and elliptic curves are reconstructed from their perfect derived categories +using triangular spectra. +This paper is organized as follows. In Section 2, we give the definitions of the +center and the triangular spectra of a triangulated category, which play a central +role throughout this paper. In Section 3, we prove Theorem 1.1 and give its ap- +plications, including the Bondal-Orlov-Ballard reconstruction theorem. In Section +4, we introduce the structure sheaf on the triangular spectrum and prove Theorem +1.2. We apply this result to the perfect derived categories of noetherian schemes +and deduce Theorem 1.3. +2. Preliminaries +In this section, we recall basic definitions and properties for later use. We begin +with our convention. +Convention 2.1. (1) Throughout this paper, we assume that all triangulated cat- +egories are essentially small and that all subcategories are full. +Let T be a triangulated category. A thick subcategory of T is a subcategory +closed under direct summands, shifts, and extensions. The set Th(T ) of all +thick subcategories forms a lattice with respect to the inclusion relation. For +M ∈ T , denote by ⟨M⟩ the smallest thick subcategory of T containing M. + +4 +HIROKI MATSUI +(2) For a noetherian scheme X, a complex F of OX-modules is said to be perfect +if, for any x ∈ X, there is an open neighborhood U ⊆ X of x such that the +restriction F|U is quasi-isomorphic to a bounded complex of locally free sheaves +of finite rank. Denote by Dpf(X) the derived category of perfect complexes on +X. We call it the perfect derived category of X. +2.1. Center of triangulated categories. We recall the definition and basic prop- +erties of the center of a triangulated category; see [16] for details. +Definition 2.2. Let T be a triangulated category. +(1) The center Z(T ) of T is the set of natural transformations η : idT → idT +with η[1] = [1]η. The composition of natural transformations makes Z(T ) a +commutative ring. +(2) We say that an element η ∈ Z(T ) is locally nilpotent if ηM is a nilpotent element +of the endomorphism ring EndT (M) for each M ∈ T . We shall denote by Z(T )lnil +the ideal of Z(T ) consisting of locally nilpotent elements and by Z(T )lred := +Z(T )/Z(T )lnil the quotient ring. +Let Φ : T → T ′ be an exact functor between triangulated categories. It seems +to be not known whether Φ induces a morphism between Z(T )lred and Z(T ′)lred in +general. Let us give several functoriality results of Z(−)lred under certain functors +following [16]. +We say that an exact functor Φ : T → T ′ is dense if, for any M′ ∈ T ′, there are +M ∈ T and N′ ∈ T ′ such that Φ(M) ∼= M′ ⊕ N′. For example, the idempotent +completion functor ι : T → T ♮ of T is fully faithful and dense; see [8]. +Lemma 2.3. Let Φ : T → T ′ be a fully faithful dense exact functor. Then there +is an isomorphism Φ∗ : Z(T ′) +∼ +=→ Z(T ), where Φ∗(η)M = Φ−1(ηΦ(M)) for M ∈ T . +Moreover, this isomorphism induces an isomorphism Φ∗ : Z(T ′)lred +∼ +=→ Z(T )lred. +Proof. We note that for any object M′ ∈ T ′, there is an object M ∈ T and an +isomorphism M′ ⊕ M′[1] ∼= Φ(M); see [3, (3.2) in the proof of Proposition 3.13]. +As Φ : T → T ′ is a fully faithful exact functor, it induces a homomorphism Φ∗ : +Z(T ′) → Z(T ) by [16, Proposition 2.3(1)]. First, we prove that this homomorphism +is an isomorphism. Let η ∈ Z(T ′) with Φ∗(η) = 0. For any M′ ∈ T ′, take an +object M ∈ T and an isomorphism M′ ⊕M′[1] ∼= Φ(M). Then we obtain Φ∗(η)M = +Φ−1(ηΦ(M)) ∼= Φ−1(ηM′)⊕Φ−1(ηM′)[1]. Therefore, Φ∗(η)M = 0 implies Φ−1(ηM′) = 0 +and hence ηM′ = 0. This shows that Φ∗ : Z(T ′) → Z(T ) is injective. To show that +Φ∗ : Z(T ′) → Z(T ) is surjective, fix an element δ ∈ Z(T ) and construct η ∈ Z(T ′) +with Φ∗(η) = δ. For any M′ ∈ T ′, take an object M ∈ T and an isomorphism +ϕ : Φ(M) +∼ +=−→ M′ ⊕ M′[1]. Then we define the morphism ηM′ : M′ → M′ as the +composition +M′ ( 1 +0) +→ M′ ⊕ M′[1] +ϕ−1 +−−→ Φ(M) +Φ(δM ) +−−−→ Φ(M) +ϕ−→ M′ ⊕ M′[1] +( 1 0 ) +→ M′. +This morphism ηM′ does not depend on the choices of M and ϕ. Indeed, take an +object N ∈ T and an isomorphism ψ : Φ(N) +∼ +=−→ M′ ⊕ M′[1]. Since Φ is full, there is + +TRIANGULAR SPECTRA AND DERIVED CATEGORIES +5 +a morphism f : M → N in T such that Φ(f) = ψ−1ϕ. Then we have the equalities +ϕΦ(δM)ϕ−1 = ψΦ(f)Φ(δM)ϕ−1 = ψΦ(δN)Φ(f)ϕ−1 = ψΦ(δN)ψ−1. +For this reason, ηM′ is well-defined. For a morphism g : M′ → N′ in T ′, we prove +gηM′ = ηN′g. Take objects M, N ∈ T and isomorphisms ϕ : Φ(M) +∼ +=−→ M′ ⊕ M′[1], +ψ : Φ(N) +∼ +=−→ N′ ⊕ N′[1] in T ′. Since Φ is full, there is a morphism f : M → N such +that Φ(f) = ψ−1 � +g +0 +0 g[1] +� +ϕ. Then we get the equalities +gηM′ = g ( 1 0 ) ϕΦ(δM)ϕ−1 ( 1 +0 ) += ( 1 0 ) +� +g +0 +0 g[1] +� +ϕΦ(δM)ϕ−1 ( 1 +0 ) += ( 1 0 ) ψΦ(f)Φ(δM)ϕ−1 ( 1 +0 ) += ( 1 0 ) ψΦ(δN)Φ(f)ϕ−1 ( 1 +0 ) += ( 1 0 ) ψΦ(δN)ψ−1 � +g +0 +0 g[1] +� +( 1 +0 ) += ( 1 0 ) ψΦ(δN)ψ−1 ( 1 +0 ) g = ηN′g. +This shows that η : idT ′ → idT ′ is a natural transformation. Moreover, one can +easily see from the definition that η[1] = [1]η holds and hence η ∈ Z(T ′). Finally, we +will check the equality Φ∗(η) = δ. For an object M ∈ T , we can take the canonical +isomorphism Φ(M ⊕ M[1]) ∼= Φ(M) ⊕ Φ(M)[1]. Using this isomorphism, we have a +commutative diagram +Φ(M) ( 1 +0) � +Φ(δM ) +� +Φ(M) ⊕ Φ(M)[1] +∼= +� Φ(δM ) +0 +0 +Φ(δM )[1] +� +� +Φ(M ⊕ M[1]) +Φ(δM⊕M[1]) +� +Φ(M) +Φ(M) ⊕ Φ(M)[1] +( 1 0 ) +� +∼= +Φ(M ⊕ M[1]) +and this means that ηΦ(M) = Φ(δM). Therefore, Φ∗(η)M = Φ−1(ηΦ(M)) = δM. Thus, +we conclude that Φ∗(η) = δ. +We finish the proof by checking that the first isomorphism induces the second one. +From the first isomorphism, the induced homomorphism Φ∗ : Z(T ′)lred ։ Z(T )lred is +surjective. Pick an element η ∈ Z(T ′) with Φ∗(η) ∈ Z(T )lnil. For any object M ∈ T , +there is an integer n > 0 such that Φ−1(ηn +Φ(M)) = Φ−1(ηΦ(M))n = Φ∗(η)n +M = 0 as +Φ∗(η) is locally nilpotent. Hence ηn +Φ(M) = 0. Because each object M′ ∈ T ′ is a direct +summand of Φ(M) for some M ∈ T , we get η ∈ Z(T ′)lred. This shows the injectivity +of Φ∗ : Z(T ′)lred ։ Z(T )lred. +■ +For thick subcategories U ⊆ V of T , there is a unique exact functor QV/U : T /U → +T /V such that QV/U ◦ QU = QV, where QU : T → T /U and QV : T → T /V are the +canonical functors. +Lemma 2.4. Let U ⊆ V be thick subcategories of T . Then the canonical functor +QV/U : T /U → T /V induces a ring homomorphism (QV/U)∗ : Z(T /U) → Z(T /V) + +6 +HIROKI MATSUI +where (QV/U)∗(η)QV(M) = QV/U(ηQU(M)) for M ∈ T . Moreover, this homomorphism +induces a homomorphism (QV/U)∗ : Z(T /U)lred → Z(T /V)lred. +Proof. It follows from [23, Proposition 2.3.1] that V/U is a thick subcategory of T /U +and that QV/U induces a triangle equivalence +Φ : (T /U)/(V/U) +∼ +=−→ T /V +such that Φ ◦ Q = QV/U, where Q : T /U → (T /U)/(V/U) is the canonical functor. +By Lemma 2.3 and [16, Proposition 2.3(2)], we obtain the homomorphism +(QV/U)∗ : Z(T /U) +Q∗ +−→ Z((T /U)/(V/U)) +(Φ∗)−1 +−−−−→ +∼ += +Z(T /V). +The second statement is clear from the description (QV/U)∗(η)QV(M) = QV/U(ηQU(M)). +■ +2.2. Spectra of triangulated categories. In this subsection, let us recall the two +kinds of spectra introduced by Balmer [3] and Matsui [17]. +Let T be a triangulated category. We set +Z(E) := {X ∈ Th(T ) | X ∩ E = ∅} +for a subcategory E of T . Then we can easily see the following properties: +(i) Z(T ) = ∅ and Z(∅) = Th(T ). +(ii) � +i∈I Z(Ei) = Z(� +i∈I Ei). +(iii) Z(E) ∪ Z(E′) = Z(E ⊕ E′), where E ⊕ E′ := {M ⊕ M′ | M ∈ E, M′ ∈ E′}. +Therefore, we can define the topology on Th(T ) whose closed subsets are of the +form Z(E) for some E ⊆ T . +A tensor triangulated category is a triple (T , ⊗, 1) consisting of a triangulated cat- +egory T together with a symmetric monoidal structure (⊗, 1) such that the bifunctor +⊗ : T × T → T is exact in each variable. +Definition 2.5. ([3, Definition 2.1]). Let (T , ⊗, 1) be a tensor triangulated category. +A proper thick subcategory P ⊊ T is said to be a prime thick ideal if it satisfies the +following conditions: +• (ideal) M ⊗ N ∈ P holds for any M ∈ T and N ∈ P, +• (prime) M ⊗ N ∈ P implies M ∈ P or N ∈ P. +Denote by Spec⊗(T ) the set of prime thick ideals of T together with the induced +topology of Th(T ). We call Spec⊗(T ) the Balmer spectrum or the tensor-triangular +spectrum of (T , ⊗, 1). +Balmer [3] also defined the structure sheaf on Spec⊗(T ). +For an open subset +U ⊆ Spec⊗(T ), set T U := � +P∈U P. We define the tensor triangulated category +T (U) by +T (U) := +� +T /T U�♮ . +Denote by 1U the image of 1 under the canonical functor T → T (U). + +TRIANGULAR SPECTRA AND DERIVED CATEGORIES +7 +Definition 2.6. ([3, Definition 6.1]). We denote by O⊗,T the sheafification of the +presheaf Op +⊗,T of commutative rings given by the assignment +U �→ Op +⊗,T (U) := EndT (U)(1U). +We simply write Spec⊗(T ) for the ringed space (Spec⊗(T ), O⊗,T ). +Using the Balmer spectrum, Balmer [3] proved that any noetherian scheme X +could be reconstructed from the tensor triangulated category (Dpf(X), ⊗L +OX, OX). +Theorem 2.7. ([3, Theorem 6.3(a)]). +For a noetherian scheme X, there is an +isomorphism +SX : X +∼ +=−→ Spec⊗(Dpf(X)) +of ringed spaces, where the map of underlying topological spaces is given by +x �→ SX(x) := {F ∈ Dpf(X) | Fx ∼= 0 in Dpf(OX,x)}. +Next, let us recall the triangular spectrum of a triangulated category of T . +Definition 2.8. ([17, Definitions 2.2 and 2.4]). Let T be a triangulated category. A +proper thick subcategory P ⊊ T is called a prime thick subcategory if the partially +ordered set {X ∈ Th(T ) | P ⊊ X } has the smallest element. Denote by Spec△(T ) +the set of prime thick subcategories of T together with the induced topology of +Th(T ). We call Spec△(T ) the triangular spectrum of T . +Remark 2.9. The above definition of a prime thick subcategory differs from that +in [17]. There, we call P a prime thick subcategory if {X ∈ Th(T ) | P ⊊ X } has a +unique minimal element. The correct definition is the one in this paper, and all the +proofs in [17] are done using the present definition (“unique minimal element” in +Lemma 2.15 and Proposition 4.7 should be changed to “smallest element”). Also, +there are no problems in the results in [12] if we adopt the definition in this paper. +Let Φ : T → T ′ be an exact functor. Since Φ−1(X ) := {M ∈ T | Φ(M) ∈ T ′} +is a thick subcategory of T for each thick subcategory X ⊆ T ′, we have an order- +preserving map +Φ∗ : Th(T ′) → Th(T ), +X �→ Φ−1(X ). +This map restricts to continuous maps between triangular spectra for fully faithful +dense exact functors and quotient functors. +Lemma 2.10. ([17, Proposition 2.11]). Let Φ : T → T ′ be a fully faithful dense +exact functor. Then the map Φ∗ : Th(T ′) → Th(T ) restricts to a homeomorphism +Φ∗ : Spec△(T ′) +∼ +=→ Spec△(T ). +Lemma 2.11. ([17, Proposition 2.9]). Let T be a triangulated category and U be its +thick subcategory. Denote by Q : T → T /K the canonical functor. Then the map +Q∗ : Th(T /K) → Th(T ) restricts to an immersion +Q∗ : Spec△(T /K) ֒→ Spec△(T ). +of topological spaces whose image is {P ∈ Spec△(T ) | K ⊆ P}. + +8 +HIROKI MATSUI +The following theorem is one of the main results in [17], which is some sort of a +generalization of Theorem 2.7. +Theorem 2.12. Let X be a noetherian scheme. +(1) Let P be a thick ideal of Dpf(X). Then P is a prime thick subcategory if and +only if P is a prime thick ideal if and only if P = SX(x) for some x ∈ X. +(2) We have an immersion +SX : X ֒→ Spec△(Dpf(X)), +x �→ SX(x). +of topological spaces whose image is Spec⊗(Dpf(X)). +Remark 2.13. We will see in Corollary 4.7 that the above immersion SX : X ֒→ +Spec△(Dpf(X)) follows from more general result Theorem 4.6. +From the definition, it is quite difficult to determine the topological space +Spec△(Dpf(X)) for a given noetherian scheme X. For special cases, triangular spec- +tra are determined as follows. +Proposition 2.14. (1) ([17, Corollary 2.17(1)]). If X is a quasi-affine noetherian +scheme, then there is a homeomorphism +Spec△(Dpf(X)) ∼= X. +(2) ([17, Example 4.10]). Let P1 be the projective line over a field. Then there is a +homeomorphism +Spec△(Dpf(P1)) ∼= P1 ⊔ Z, +where Z is considered as the discrete topological space. +(3) ([12, Theorem 4.11]). Let E be an elliptic curve over a field. Then there is a +homeomorphism +Spec△(Dpf(E)) ∼= E ⊔ + + � +(r,d)∈I +Er,d + + . +Here, I := {(r, d) ∈ Z2 | r > 0, gcd(r, d) = 1} and Er,d is a copy of E for each +(r, d) ∈ I. +3. Reconstruction of schemes +In this section, we discuss the reconstruction of a noetherian scheme X from +the triangulated category Dpf(X). Reconstruction of underlying topological spaces +has been discussed in [12, 17] in terms of the triangular spectrum of Dpf(X). To +reconstruct the structure sheaf, centers of triangulated categories play a crucial role. +We recall that a tensor triangulated category (T , ⊗, 1) is rigid if it is closed, i.e., +the exact functor M ⊗ − : T → T has a right adjoint [M, −] : T → T for each +M ∈ T and such that every object M is strongly dualizable, i.e., the canonical map +[M, 1] ⊗ N → [M, N] +is an isomorphism for each N ∈ T ; see [13] for details. If T is rigid, then so is T (U) +for any open subset U ⊆ Spec⊗(T ) by [4, Proposition 2.15]. + +TRIANGULAR SPECTRA AND DERIVED CATEGORIES +9 +We say that the tensor triangulated category (T , ⊗, 1) is monogenic if T is gener- +ated by 1, i.e., T = ⟨1⟩. We say that T is locally monogenic if, for any prime thick +ideal P ∈ Spec⊗(T ), there is a quasi-compact open neighborhood U ⊆ Spec⊗(T ) of +P such that T (U) is monogenic. +We begin with stating the following general result for specific tensor triangulated +categories whose proof is taken from [19, Lemma 4.10]. +Proposition 3.1. Let (T , ⊗, 1) be an idempotent complete, rigid, locally monogenic +tensor triangulated category. Then the evaluation at 1 induces an isomorphism +Z(T )lred ∼= EndT (1)red. +Proof. As Spec⊗(T ) is a spectral space, there are quasi-compact open covering +U1, U2, . . . , Un of Spec⊗(T ) such that T (Ui) = ⟨1Ui⟩ for i = 1, 2, . . . , n. +Let α : Z(T ) → EndT (1), η �→ η1 be the evaluation at 1. Since α has a right +inverse EndT (1) → Z(T ), φ �→ φ ⊗ (−), the homomorphism α is surjective. There- +fore, it suffices to show that the induced homomorphism α : Z(T )lred → EndT (1)red +is injective. To this end, let us prove η ∈ Z(T )lnil for each η ∈ Z(T ) with α(η) = 0, +i.e., η1 = 0. We proceed by induction on n. +First assume n = 1, i.e., T = ⟨1⟩. We prove that ηM is nilpotent for any M ∈ T . +We consider the subcategory X ⊆ T consisting of objects M ∈ T with ηM nilpotent. +Then X contains 1 as η1 = 0. In addition, X is a thick subcategory. Indeed, it is +clear that X is closed under direct summands and shifts. Take an exact triangle +L +f−→ M +g−→ N → L[1] in T with L, N ∈ X . Then there is an integer l ≥ 1 such that +ηl +L = 0 and ηl +N = 0. From the naturality of ηl, the equality gηl +M = ηl +Ng = 0 holds. +Thus, ηl +M factors as ηl +M : M +a−→ L +f−→ M. Again using the naturality of ηl, we get +η2l +M = ηl +Mηl +M = ηl +Mfa = fηl +La = 0. As a result, M ∈ X follows. Therefore, X is +thick and so X = ⟨1⟩ = T . This means that η is locally nilpotent. +Next, assume that n > 1 and set V = U2 ∪ U3 ∪ · · ·Un. Assume further that the +evaluations at the unit objects induce isomorphisms +α : Z(T (U1))lred +∼ +=−→ EndT (U1)(1U1)red, +α : Z(T (V ))lred +∼ +=−→ EndT (V )(1V )red. +Here, we note that there is a homeomorphism f : Spec⊗(T (V )) +∼ +=−→ V such that +T (V )(f −1(Ui)) ∼= T (Ui) = ⟨1Ui⟩ for i = 2, 3, . . . , n; see [7, Proposition 1.11] and [6, +Constructions 24 and 29]. Hence we can apply the induction hypothesis to T (V ). +One can easily verify that the canonical functors Q : T +→ T /T U1 and ι : +T /T U1 → T (U1) induce a commutative diagram +Z(T )lred +Q∗ +� +α +� +Z(T /T U1)lred +α +� +Z(T (U1))lred +α +∼ += +� +ι∗ +∼ += +� +EndT (1)red +Q +� EndT /T U1(Q(1))red +ι +∼ += +� EndT (U1)(1U1)red, +where the evaluations at the unit objects induce the vertical arrows. +From this +diagram, η1 = 0 yields Q∗(η) ∈ Z(T /T U1)lnil. For an object M ∈ T , there is an + +10 +HIROKI MATSUI +integer l ≥ 1 such that Q(ηl +M) = Q∗(η)l +Q(M) = 0. Accordingly, ηl +M factors some +N ∈ T U1. Then η(d+1)l +M +factors ηdl +N for any integer d ≥ 1. +For the canonical functors Q′ : T → T /T V and ι′ : T /T V → T (V ), the same +argument as above shows that Q′ +∗(η) ∈ Z(T /T V )lnil. Here we use the isomorphism +α : Z(T (V ))lred +∼ +=−→ EndT (V )(1V )red which is our induction hypothesis. In particular, +Q′(ηdl +N) = Q′ +∗(η)dl +Q′(N) = 0 for some d ≥ 1. It follows from [7, Theorem 7.1] that the +functor Q′ restricts to a fully faithful functor +Q′ : T U1 → (T /T V )U1 +and hence Q′(ηdl +N) = 0 implies ηdl +N = 0. Then η(d+1)l +M += 0 holds as η(d+1)l +M +factors ηdl +N. +As a result, we conclude that η ∈ Z(T )lnil. +■ +Let X be a noetherian scheme and U ⊆ X be an open subset with Z := X \ U. +Then [2, Theorem 2.13] shows that the restriction functor (−)|U: Dpf(X) → Dpf(U) +induces a triangle equivalence +� +Dpf(X)/Dpf +Z (X) +�♮ ∼= Dpf(U) +of tensor triangulated categories, where Dpf +Z (X) = {F ∈ Dpf(X) | F|U∼= 0} = +� +x∈U SX(x). Therefore, the left-hand side is Dpf(X)(SX(U)). Applying Proposition +3.1 to T = Dpf(X), the following result is recovered. +Corollary 3.2. ([2, Proposition 8.1] and [19, Lemma 4.10]). +For a noetherian +scheme X, there is an isomorphism +Z(Dpf(X))lred ∼= Γ(X, OX)red. +Proof. Dpf(X) is idempotent complete because it is realized as the full subcategory +of compact objects of Dqc(Mod OX). Also, it is rigid by [4, Proposition 4.1]. For any +affine scheme U, it holds that Dpf(U) = ⟨OU⟩. It follows from Theorem 2.7 and [2, +Theorem 2.13] that the tensor triangulated category Dpf(X) is locally monogenic. +Thus, we get isomorphisms +Z(Dpf(X))lred ∼= EndDpf(X)(OX)red ∼= Γ(X, OX)red +of commutative rings from Proposition 3.1. +■ +Now we state and prove the first main result in this paper about the reconstruction +of a noetherian scheme from the perfect derived category. +Theorem 3.3. Let X1 and X2 be noetherian schemes and Φ : Dpf(X1) +∼ +=−→ Dpf(X2) +be a triangle equivalence. Assume that there is a line bundle Li on Xi for i = 1, 2 +satisfying the following conditions: +(i) Li is ample or anti-ample for i = 1, 2. +(ii) There is an isomorphism +Φ(F ⊗L +OX1 L1) ∼= Φ(F) ⊗L +OX2 L2 +for any F ∈ Dpf(X1). +Then (X1)red and (X2)red are isomorphic as schemes. + +TRIANGULAR SPECTRA AND DERIVED CATEGORIES +11 +Proof. For i = 1, 2, denote by SpecLi +△ (Dpf(Xi)) the subset of Spec△(Dpf(Xi)) con- +sisting of prime thick subcategories P with P ⊗L +OXi Li := {F ⊗L +OXi Li | F ∈ P} = P. +Then the homeomorphism Φ∗ : Spec△(Dpf(X2)) +∼ +=−→ Spec△(Dpf(X1)) restricts to the +homeomorphism Φ∗ : SpecL2 +△ (Dpf(X2)) +∼ +=−→ SpecL1 +△ (Dpf(X1)) by (ii). On the other +hand, as Li is ample or anti-ample by (i), {L⊗n +i +| n ∈ Z} generates Dpf(Xi), and +hence Theorem 2.12(1) implies SpecLi +△ (Dpf(X1)) = Spec⊗(Dpf(Xi)). From Theorem +2.7, we obtain a homeomorphism SXi : Xi +∼ +=−→ Spec⊗(Dpf(X1)) = SpecLi +△ (Dpf(Xi)). +Thus, we get a homeomorphism +f : X2 +SX2 +−−→ SpecL2 +△ (Dpf(X2)) +Φ∗ +−→ SpecL1 +△ (Dpf(X1)) +S−1 +X1 +−−→ X1. +The construction of this map yields Φ(SX1(f(x2))) = SX2(x2) for any x2 ∈ X2. +For an open subset U ⊆ X2 with Z = X2 \ U, one has +Φ(Dpf +f (Z )(X1)) = Φ + + +� +x1∈f(Z) +SX1(x1) + + = +� +x1∈f(Z) +Φ (SX1(x1)) = +� +x2∈Z +SX2(x2) = Dpf +Z (X2). +Therefore, the triangle equivalence Φ : Dpf(X1) +∼ +=−→ Dpf(X2) induces a triangle equiv- +alence +Φ : Dpf(X1)/Dpf +f (Z )(X1) +∼ +=−→ Dpf(X2)/Dpf +Z (X2). +Taking idempotent completion and Z(−)lred, we obtain an isomorphism +Γ(f(U), OX1)red ∼= Γ(U, OX2)red +by [2, Theorem 2.13] and Corollary 3.2. As we can easily see that this isomorphism +is compatible with the restriction of open subsets, we get an isomorphism (X2)red ∼= +(X1)red of schemes. +■ +This result has several applications. The first one is the following reconstruction +theorem which is well-known at least for the affine case. +Corollary 3.4. Let X1 and X2 be noetherian quasi-affine schemes. If there is a tri- +angle equivalence Φ : Dpf(X1) +∼ +=−→ Dpf(X2), then (X1)red and (X2)red are isomorphic +as schemes. +Proof. Take Li to be OXi. Then OXi is an ample line bundle as Xi is quasi-affine. +Moreover, the isomorphism +Φ(F ⊗L +OX1 OX1) ∼= Φ(F) ⊗L +OX2 OX2 +obviously holds for each F ∈ Dpf(X1); hence, the result follows by Theorem 3.3. +■ +Another application of Theorem 3.3 is to recover the famous result by Bondal- +Orlov [9] and Ballard [1]. +Corollary 3.5. ([1, 9]). Let Xi be a Gorenstein projective scheme over a field k with +ample or anti-ample canonical bundle ωXi for i = 1, 2. If there is a k-linear triangle + +12 +HIROKI MATSUI +equivalence Φ : Dpf(X1) +∼ +=−→ Dpf(X2), then (X1)red and (X2)red are isomorphic as +schemes. +Proof. Take Li to be ωXi. Then the assumption (i) is satisfied. Recall that a Serre +functor SXi : Dpf(Xi) → Dpf(Xi) on Dpf(Xi) is a triangle equivalence such that there +is a natural isomorphism +HomDpf(Xi)(F, G) ∼= HomDpf(Xi)(G, SXi(F))∗ +for any F, G ∈ Dpf(Xi). Here (−)∗ stands for the k-dual functor. It follows from +[1, Lemma 6.6] that SXi ∼= (−) ⊗L +OXi ωXi[− dim Xi] and from [14, Lemma 1.30] that +SX2 ◦ Φ ∼= Φ ◦ SX1. Therefore, the assumption (ii) follows. Applying Theorem 3.3, +we get an isomorphism between (X1)red and (X2)red. +■ +Remark 3.6. (1) The original statement by Bondal-Orlov [9] and Ballard [1] as- +sumes that either ωX1 or ωX2 is ample or anti-ample. Although Corollary 3.5 +requires both of them to be ample or anti-ample, our argument is purely alge- +braic. +(2) There is a generalization of the Bondal-Orlov-Ballard reconstruction theorem to +the relative case by Sancho de Salas and Sancho de Salas [20]. Theorem 3.3 also +recovers their result by the same argument as in Corollary 3.5. +4. Triangular spectra as ringed spaces +One of the key ingredients of proof of Theorem 3.3 is the isomorphisms +Z + + +� +Dpf(X)/ +� +x∈U +SX(x) +�♮ + +lred +∼= Z +� +Dpf(U) +� +lred ∼= Γ(U, OX)red +for an open subset U ⊆ X, which is a consequence of [2, Theorem 2.13] and Corol- +lary 3.2. Motivated by these isomorphisms, we define the structure sheaf on the +triangular spectrum for a triangulated category. +Let T be a triangulated category. For an open subset U ⊆ Spec△(T ), we set +T U := +� +P∈U +P, +T (U) := +� +T /T U�♮ . +By composing the quotient functor QU : T → T /T U and the idempotent completion +functor ιU : T /T U → T (U), we have a natural functor resU : T → T (U). Thanks +to Lemmas 2.3 and 2.4, the functor resU induces a homomorphism +(resU)∗ : Z(T )lred +(QU)∗ +−−−→ Z(T /T U)lred +((ιU )∗)−1 +−−−−−→ +∼ += +Z(T (U))lred. +Let U ⊆ V ⊆ Spec△(T ) be open subsets. Then the universal properties of the +Verdier quotient and the idempotent completion shows that the inclusion T V ⊆ T U +induces a unique exact functor +resV,U : T (V ) → T (U) + +TRIANGULAR SPECTRA AND DERIVED CATEGORIES +13 +such that resV,U ◦resV = resU. Again using Lemmas 2.3 and 2.4, this functor induces +a homomorphism +(resV,U)∗ : Z(T (V ))lred → Z(T (U))lred. +Furthermore, we have the equality (resW,V )∗ ◦ (resV,U)∗ = (resW,U)∗ for three open +subsets U ⊆ V ⊆ W ⊆ Spec△(T ). +Definition 4.1. Let T be a triangulated category. We define the sheaf O△,T of +commutative rings on Spec△(T ) by the sheafification of the presheaf Op +△,T : +Op +△,T (U) := Z(T (U))lred, +ρV,U := (resV,U)∗ : Op +△,T (V ) → Op +△,T (U). +Later, we consider the spectrum Spec△(T ) as the ringed space (Spec△(T ), O△,T ). +Recall that a morphism (f, f ♭) : (X, OX) → (Y, OY ) of ringed spaces consists of +a continuous map f : X → Y and a morphism f ♭ : f −1OY → OX of sheaves of +commutative rings on X. We say that the morphism (f, f ♭) is an open immersion +of ringed spaces if f : X → Y is an open immersion of topological spaces and +f ♭ : f −1OY → OX is an isomorphism. +We remark that for a continuous map +f : X → Spec△(T ), the pullback sheaf f −1O△,T is isomorphic to the sheafification +of the presheaf +f pOp +△,T : U �→ +lim +−→ +f(U)⊆V +Op +△,T (V ) +of X. +Lemmas 2.3 and 2.4 show that the continuous maps in Lemmas 2.10 and 2.11 can +be extended to morphisms of ringed spaces. +Proposition 4.2. Let Φ : T → T ′ be a fully faithful dense exact functor. Then we +have an isomorphism +Φ∗ : Spec△(T ′) +∼ +=−→ Spec△(T ) +of ringed spaces. +Proof. This follows from 2.3 and Lemmas 2.10. +■ +Proposition 4.3. Let U be a thick subcategory of T . Then the quotient functor +Φ : T → T /U induces a morphism +Φ∗ : Spec△(T /K) → Spec△(T ) +of ringed spaces. Moreover, if the image of Φ∗ is an open subset of Spec△(T ), then +the above morphism is an open immersion of ringed spaces. +Proof. We construct a morphism (Φ∗)♭ : (Φ∗)−1O△,T → O△,T /K of sheaves of com- +mutative rings. +Take open subsets U ⊆ Spec△(T /K) and V ⊆ Spec△(T ) with +Φ∗(U) ⊆ V . +Setting �U := Φ∗(U) = {P ∈ Spec△(T ) | P/K ∈ U}, one sees +(T /K)U = � +P∈ �U P/K = T �U/K. Then the inclusion T V ⊆ T �U induces a triangle +functor +T /T V → T /T +�U ∼= (T /K)/(T +�U/K) = (T /K)/(T /K)U, +where the triangle equivalence is by [23, Proposition 2.3.1(c)]. Applying Z((−)♮), we +get a homomorphism Op +△,T (V ) → Op +△,T /K(U). Moreover, let U ⊆ U′ ⊆ Spec△(T /K) + +14 +HIROKI MATSUI +and V ⊆ V ′ ⊆ Spec△(T ) be other open subsets with � +U′ := Φ∗(U′) ⊆ V ′. Then the +inclusions +T V � � +� T �U +T V ′ � � +� +�� +� +T +� +U′ +�� +� +induce a commutative diagram +T /T V +� T /T �U +∼= +(T /K)/(T /T �U) +T /T V ′ +� +� +T /T +� +U′ +� +∼= +(T /K)/(T /T +� +U′). +� +Applying Z((−)♮)lred, we obtain a commutative diagram +Op +△,T (V ) +� Op +△,T /K(U) +Op +△,T (V ′) +� +� +Op +△,T /K(U′). +� +Therefore, +the homomorphism Op +△,T (V ) +→ +Op +△,T /K(U) defines a morphism +(Φ∗)pOp +△,T → Op +△,T /K of presheaves of commutative rings. Taking sheafifications, +we obtain a morphism (Φ∗)♭ : (Φ∗)−1O△,T → O△,T /K of sheaves of commutative +rings. +Next, assume Φ∗(Spec△(T /K)) ⊆ Spec△(T ) is an open subset. +Then Φ∗ : +Spec△(T /K) → Spec△(T ) is an open immersion of topological spaces. +For any +open subset U ⊆ Spec△(T /K), the map (Φ∗)pOp +△,T /K(U) → Op +△,T (U) is induced +from the triangle equivalence +T /T +�U ∼= (T /K)/(T +�U/K) = (T /K)/(T /K)U. +As a result, the map (Φ∗)pOp +△,T /K → Op +△,T is an isomorphism and hence so is +(Φ∗)♭ : (Φ∗)−1O△,T /K → O△,T . +■ +Corollary 4.4. For an object M ∈ T , the quotient functor Φ : T → T /⟨M⟩ induces +an open immersion +Φ∗ : Spec△(T /⟨M⟩) ֒→ Spec△(T ) +of ringed spaces. +Proof. The image of Φ∗ is +{P ∈ Spec△(T ) | M ∈ T } = Spec△(T ) \ Z({M}), +which is an open subset of Spec△(T ). The result follows from Proposition 4.3. +■ +The main theorem in this section compares the Balmer spectrum and the tri- +angular spectrum for an idempotent complete, rigid, and locally monogenic tensor +triangulated category. We treat the monogenic case first. + +TRIANGULAR SPECTRA AND DERIVED CATEGORIES +15 +Lemma 4.5. Let T be an idempotent complete, rigid, monogenic tensor triangu- +lated category. +Assume further that Spec⊗(T ) is noetherian. +Then the equality +Spec△(T ) = Spec⊗(T ) holds. +Proof. Since T = ⟨1⟩, every thick subcategory is a thick ideal. Therefore, by [3, +Remark 4.3], the thick subcategories and the radical thick ideals coincide. +Let P ∈ Spec△(T ) and take the smallest element X in {X ∈ Th(T ) | P ⊊ X }. +Since P is a radical thick ideal by the above argument, it is the intersection of all +prime thick ideals Q containing P by [3, Lemma 4.2]. If P is not a prime thick ideal, +then such Q satisfies P ⊊ Q and hence X ⊆ Q by the assumption on X . Therefore, +P ⊊ X ⊆ � +Q∈Spec⊗(T ),P⊆Q Q = P leads a contradiction. Hence we conclude that +the inclusion Spec△(T ) ⊆ Spec⊗(T ) holds. +Conversely, take P ∈ Spec⊗(T ) and prove P ∈ Spec△(T ). It follows from [3, The- +orem 4.10] that there is an order-preserving bijection between the thick subcategories +of T and the specialization-closed subsets of Spec⊗(T ). The noetherian assumption +is used here. Under this bijection, P corresponds to the specialization-closed subset +W := {Q ∈ Spec⊗(T ) | P ̸⊆ Q} = {Q ∈ Spec⊗(T ) | P ̸∈ {Q}}. +Here we use [3, Proposition 2.9]. Therefore, it suffices to show that there is the +smallest specialization-closed subset T with W ⊊ T. +Set T := W ∪ {P} and +prove that this is the specialization-closed subset that we need. For an element +Q ∈ {P} \ {P}, Q belongs to W as P ̸⊆ Q. +Thus T = W ∪ {P} holds and +this is specialization-closed. For a specialization-closed subset W ′ ⊆ Spec⊗(T ) with +W ⊊ W ′, we will check T ⊆ W ′. Since W ⊊ W ′, there is an element Q ∈ W ′ such +that Q ̸∈ W. Then Q ̸∈ W implies P ∈ {Q}. As W ′ is specialization-closed, one +has P ∈ {Q} ⊆ W ′. This shows that the inclusion T ⊆ W ′ and hence T is the +smallest specialization-closed subset with W ⊊ T. +■ +Theorem 4.6. Let (T , ⊗, 1) be an idempotent complete, rigid, locally monogenic +tensor triangulated category. Assume further that Spec⊗(T ) is noetherian. Then +there is a morphism +i : Spec⊗(T )red → Spec△(T ) +of ringed spaces satisfying the following conditions: +(1) The map of underlying topological spaces is the inclusion. In particular, every +prime thick ideal is a prime thick subcategory. +(2) i is an open immersion of ringed spaces whenever Spec⊗(T ) is an open subset +of Spec△(T ). +(3) i is an isomorphism of ringed spaces if T is monogenic. +Proof. First, we prove (1). Let P ∈ Spec⊗(T ). Take a quasi-compact open subset +P ∈ U ⊆ Spec⊗(T ) with T (U) = ⟨1U⟩. Then T (U) is also idempotent complete, +rigid, and locally monogenic. On the other hand, it follows from [7, Proposition 1.11] +that the canonical functor resU : T → T (U) induces a homeomorphism (resU)∗ : +Spec⊗(T (U)) +∼ +=−→ U. In particular, Spec⊗(T (U)) is noetherian. Applying Lemma + +16 +HIROKI MATSUI +4.5 yields that Spec⊗(T (U)) = Spec⊗(T (U)). Then we obtain the inclusion U ⊆ +Spec△(T (U)) as the composition +U +((resU)∗)−1 +−−−−−−→ Spec⊗(T (U)) = Spec△(T (U)) +(resU)∗ +−−−−→ Spec△(T ). +Therefore, P ∈ U ⊆ Spec△(T (U)) and hence we get Spec⊗(T ) ⊆ Spec△(T ). +Next, let us construct a morphism i−1O△,T → (O⊗,T )red. For open subsets U ⊆ +Spec⊗(T ) and V ⊆ Spec△(T ) with U ⊆ V , the inclusion T V ⊆ T U induces a +homomorphism +Op +△,T (V ) = Z (T (V ))lred → Z (T (U))lred ∼= EndT (U)(1U)red = Op +⊗,T (U)red, +where the isomorphism is proved in Proposition 3.1. +Moreover, for other open +subsets U ⊆ U′ ⊆ Spec⊗(T ) and V ⊆ V ′ ⊆ Spec△(T ) with U′ ⊆ V ′, we get the +commutative diagram +T /T V +� T /T U +T /T V ′ +� +� +T /T U′. +� +Applying Z((−)♮)lred yields a commutative diagram +Op +△,T (V ) +Z(T (V ))lred +� Z(T (U))lred +∼= +Op +⊗,T (U)red +Op +△,T (V ′) +� +Z(T (V ′))lred +� +� +Z(T (U′))lred +� +∼= +Op +⊗,T (U′)red. +� +Therefore, Op +△,T (V ) → Op +⊗,T (U)red defines a morphism ipOp +△,T → (Op +⊗,T )red of +presheaves of commutative rings. +Taking sheafifications, we obtain a morphism +i♭ : i−1O△,T → (O⊗,T )red of sheaves of commutative rings. +Assume Spec⊗(T ) is an open subset of Spec△(T ). Then for any open subset U ⊆ +Spec⊗(T ), the homomorphism ipOp +△,T (U) → Op +⊗,T (U) is given by the isomorphism +Z(T (U))lred ∼= EndT (U)(1U)red +in Proposition 3.1. +Thus, ipOp +△,T (U) → Op +⊗,T (U) is an isomorphism and hence +so is i−1O△,T (U) → O⊗,T (U). This means that i : Spec⊗(T ) → Spec△(T ) is an +open immersion of ringed spaces. In particular, i : Spec⊗(T ) → Spec△(T ) is an +isomorphism of ringed spaces if T is monogenic. +■ +As Dpf(X) is an idempotent complete, rigid, locally monogenic tensor triangu- +lated category, we get the following immediate consequence of the combination of +Theorems 2.7 and 4.6. +Corollary 4.7. For a noetherian scheme X, there is a morphism of ringed spaces +SX : Xred → Spec△(Dpf(X)), +x �→ SX(x), +which is an open immersion of ringed spaces if SX(X) is open in Spec△(Dpf(X)). +In particular, SX : Xred → Spec△(Dpf(X)) is an isomorphism of ringed spaces if X +is quasi-affine. + +TRIANGULAR SPECTRA AND DERIVED CATEGORIES +17 +Remark 4.8. Since the ringed space Spec△(Dpf(X)) is completely determined by +the triangulated category structure on Dpf(X), Corollary 4.7 implies Corollary 3.4. +Moreover, using Corollary 4.7, we determine Spec△(Dpf(X)) for X which appeared +in Proposition 2.14. +Corollary 4.9. Let P1 be the projective line over a field k. Then there is an iso- +morphism +Spec△(Dpf(P1)) ∼= P1 ⊔ +�� +n∈Z +Spec(k) +� +of ringed spaces. +Proof. For any integer n ∈ Z, it follows from [14, Corollary 8.29] that we get triangle +equivalences Dpf(P1)/⟨OP1(n)⟩ ∼= ⟨OP1(n + 1)⟩ ∼= Dpf(Spec(k)). +It follows from +Corollaries 4.4 and 4.7 that there is an open immersion +fn : Spec(k) +SSpec(k) +−−−−→ +∼ += +Spec△(Dpf(Spec(k))) ∼= Spec△(Dpf(P1)/⟨OP1(n)⟩) ֒→ Spec△(Dpf(P1)) +of ringed spaces whose image is ⟨OP1(n)⟩. Moreover, Corollary 4.7 and [12, Corollary +4.7] show that there is an open immersion +g := SP1 : P1 ֒→ Spec△(Dpf(P1)) +of ringed spaces. +As it is proved in [17, Example 4.10], Spec△(Dpf(P1)) is the +disjoint union of the images of fn (n ∈ Z) and g. Therefore, fn (n ∈ Z) and g +induce isomorphism +P1 ⊔ +�� +n∈Z +Spec(k) +� +∼= Spec△(Dpf(P1)) +of ringed spaces. +■ +Corollary 4.10. Let E be an elliptic curve over an algebraically closed field. Then +there is an isomorphism +Spec△(Dpf(E)) ∼= E ⊔ + + � +(r,d)∈I +Er,d + + +of ringed spaces. Here, I := {(r, d) ∈ Z2 | r > 0, gcd(r, d) = 1} and Er,d is a copy of +E for each (r, d) ∈ I. +Proof. For an element (r, d) ∈ I, M(r, d) denotes the moduli space of µ-semistable +sheaves with Chern character (r, d). It follows from [22, Theorem 1] that M(r, d) ∼= +Er,d, where Er,d is a copy of E. Moreover, [11, Proposition 3] shows that there is a +triangle equivalence +Φr,d : Dpf(E) +∼ +=−→ Dpf(M(r, d)). +Then we get open immersions +fr,d : Er,d ∼= M(r, d) +SM(r,d) +֒→ +Spec△(Dpf(M(r, d))) +Φ∗ +r,d +−−→ +∼ += +Spec△(Dpf(E)), + +18 +HIROKI MATSUI +g := SE : E ֒→ Spec△(Dpf(E)) +of ringed spaces by Corollary 4.7 and [12, Corollary 4.7]. It is shown in [12, Theorem +4.11] that Spec△(Dpf(E)) is the disjoint union of open subsets fr,d(Er,d) ((r, d) ∈ I) +and g(E). Hence fr,d ((r, d) ∈ I) and g induce an isomorphism +E ⊔ + + � +(r,d)∈I +Er,d + + ∼ +=−→ Spec△(Dpf(E)) +of ringed spaces. +■ +Remark 4.11. Let E and E′ be elliptic curves over an algebraically closed field. +If there is a triangle equivalence Dpf(E) ∼= Dpf(E′), then Corollary 4.10 shows that +there is an isomorphism ⊔n∈ZE ∼= ⊔n∈ZE′. As their connected components, this +implies that E and E′ are isomorphic. This fact is well-known to experts and can be +found in [14, Pages 134 and 135] for example. Our argument is completely different +from the known argument. +References +[1] M. R. Ballard, Derived categories of sheaves on singular schemes with an application to +reconstruction, Adv. Math. 277 (2011), no. 2, 895–919. +[2] P. Balmer, Presheaves of triangulated categories and reconstruction of schemes, Math. Ann. +324 (2002), 557–580. +[3] P. Balmer, The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. +Math. 588 (2005), 149–168. +[4] P. Balmer, Supports and filtrations in algebraic geometry and modular representation theory, +Amer. J. Math., 129 (2007), no. 5, 1227–1250. +[5] P. Balmer, Spectra, spectra, spectra – Tensor triangular spectra versus Zariski spectra of +endomorphism rings, Algebr. Geom. Topol. 10 (2010), no. 3, 1521–1563. +[6] P. Balmer, Tensor triangular geometry, in Proceedings of the International Congress of Math- +ematicians, Volume II, Hindustan Book Agency, New Delhi, 2010, 85–112. +[7] P. Balmer and G. 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Press, Oxford, 2006, viii+307pp. +[15] J. Koll´ar, M. Lieblich, M. Olsson, and W. Sawin, Topological reconstruction theorems +for varieties, arXiv:2003.04847. + +TRIANGULAR SPECTRA AND DERIVED CATEGORIES +19 +[16] H. Krause and Y. Ye, On the centre of a triangulated category, Proc. Edinb. Math. Soc. +(2) 54 (2011), no. 2, 443–466. +[17] H. Matsui, Prime thick subcategories and spectra of derived and singularity categories of +noetherian schemes, Pac. J. Math. 313 (2021), no. 2, 433–457. +[18] S. Mukai, Duality between D(X) and D( ˆX) with its application to Picard sheaves, Nagoya +Math. J. 81 (1981), 153–175. +[19] R. Rouquier, Derived categories and algebraic geometry, in Triangulated categories, London +Math. Soc. Lecture Note Ser., 375, Cambridge Univ. Press, Cambridge, 2010, 351–370. +[20] C. Sancho de Salas and F. Sancho de Salas, Reconstructing schemes from the derived +category, Proc. Edinb. Math. Soc. (2) 55 (2012), no. 3, 781–796. +[21] D. Spence, Reconstruction of projective curves from the derived category, Michigan Math. J. +Advance Publication (2021), 1–24. +[22] L. W. Tu, Semistable bundles over an elliptic curve, Adv. Math. 98 (1993), no. 1, 1-26. +[23] J.-L. Verdier, Des cat´egories d´eriv´ees des cat´egories ab´eliennes, Ast´erisque, 239 (1996), +xii+253pp. +Department of Mathematical Sciences, Faculty of Science and Technology, +Tokushima University, 2-1 Minamijyousanjima-cho, Tokushima 770-8506, Japan +Email address: hmatsui@tokushima-u.ac.jp + diff --git a/VNE1T4oBgHgl3EQfbASE/content/tmp_files/load_file.txt b/VNE1T4oBgHgl3EQfbASE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a5d2085e32bfbd96024cb75054f66f7794b72b5e --- /dev/null +++ b/VNE1T4oBgHgl3EQfbASE/content/tmp_files/load_file.txt @@ -0,0 +1,753 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf,len=752 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='03168v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='AG] 9 Jan 2023 TRIANGULAR SPECTRA AND THEIR APPLICATIONS TO DERIVED CATEGORIES OF NOETHERIAN SCHEMES HIROKI MATSUI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For a triangulated category T , Matsui recently introduced a topo- logical space Spec△(T ) which we call the triangular spectrum of T as an analog of the Balmer spectrum introduced by Balmer for a tensor triangulated category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' In the present paper, we use the triangular spectrum to reconstruct a noetherian scheme X from its perfect derived category Dpf(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' As an application, we give an alternative proof of the Bondal-Orlov-Ballard reconstruction theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Moreover, we define the structure sheaf on Spec△(T ) and compare the triangular spectrum and the Balmer spectrum as ringed spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Introduction In this paper, we consider the reconstruction problem of noetherian schemes from their perfect derived categories: does a triangle equivalence between perfect derived categories Dpf(X) ∼= Dpf(Y ) imply an isomorphism X ∼= Y of noetherian schemes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Many authors have studied this kind of reconstruction problem well;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' see [1, 2, 9, 10, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' It is well-known that affine noetherian schemes are reconstructed using the triangulated category structures from their perfect derived categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' By contrast, this reconstruction problem fails for non-affine noetherian schemes in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For example, Mukai [18] proved that for an abelian variety A, A and its dual A∨ (which are not isomorphic in general) have the equivalent perfect derived categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Therefore, the triangulated category structure is insufficient for reconstructing X from Dpf(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Balmer proved that X could be reconstructed from Dpf(X) using the tensor triangulated category structure as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For an essentially small tensor triangulated category (T , ⊗, 1), Balmer defined the ringed space Spec⊗(T ), which we call the Balmer spectrum of (T , ⊗, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' In [3], he proved that there is an isomorphism X ∼= Spec⊗(Dpf(X)) for the tensor triangulated category (Dpf(X), ⊗L OX, OX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' This isomorphism shows that X is reconstructed from Dpf(X) using the tensor triangulated category struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Balmer spectra allow us to study tensor triangulated categories via algebro- geometric methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' This theory is called tensor triangular geometry and has been actively studied in various fields of mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Recently, Matsui [17] introduced the topological space Spec△(T ) for a triangulated category T to generalize tensor 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' 14A15, 14F08, 14H52, 18G80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Balmer spectrum, elliptic curve, noetherian scheme, perfect derived category, tensor triangulated category, triangular spectrum, triangulated category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' The author was partly supported by JSPS Grant-in-Aid for Early-Career Scientists 22K13894.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' 1 2 HIROKI MATSUI triangular geometry to triangulated categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' We call Spec△(T ) the triangular spectrum of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For the perfect derived category Dpf(X) of a noetherian scheme X, it is shown in [17] that there is an immersion X ֒→ Spec△(Dpf(X)) of topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' One of the most famous reconstruction results is due to Bondal-Orlov [9] and Ballard [1], which states that for Gorenstein projective varieties X1 and X2 over a field k with ample or anti-ample canonical bundles, X1 and X2 are isomorphic as varieties whenever their perfect derived categories are equivalent as k-linear trian- gulated categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' As an application of triangular spectra, we prove the following result, which generalizes the Bondal-Orlov-Ballard reconstruction theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let X1 and X2 be noetherian schemes, and Φ : Dpf(X1) ∼ =−→ Dpf(X2) be a triangle equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Assume there is a line bundle Li on Xi for i = 1, 2 satisfying the following conditions: (i) Li is ample or anti-ample for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' (ii) There is an isomorphism Φ(F ⊗L OX1 L1) ∼= Φ(F) ⊗L OX2 L2 for any F ∈ Dpf(X1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then (X1)red and (X2)red are isomorphic as schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Actually, the reconstruction of underlying topological spaces essentially appeared in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Although they use the result of Koll´ar-Lieblich-Olsson-Sawin [15] to reconstruct structure sheaves, in this paper, we give a more elementary and algebraic proof for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' To reconstruct the structure sheaf, the center of a triangulated category plays an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' So far, we consider the triangular spectrum Spec△(T ) as a topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' This paper introduces the structure sheaf on Spec△(T ) for a triangulated category T and makes Spec△(T ) the ringed space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' The following second main theorem compares the Balmer spectrum and the triangular spectrum as ringed spaces: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let (T , ⊗, 1) be an idempotent complete, rigid, and locally monogenic tensor triangulated category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Assume further that Spec⊗(T ) is noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then there is a morphism i : Spec⊗(T )red → Spec△(T ) of ringed spaces satisfying the following conditions: (1) The map of underlying topological spaces is the inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' (2) i is an open immersion of ringed spaces whenever Spec⊗(T ) is an open subset of Spec△(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' (3) i is an isomorphism of ringed spaces if T is generated by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Here, a tensor triangulated category (T , ⊗, 1) is said to be rigid if it is closed and every object is strongly dualizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' (T , ⊗, 1) is said to be is locally monogenic if it is locally generated by the unit object;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' see Section 3 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' TRIANGULAR SPECTRA AND DERIVED CATEGORIES 3 The assumptions in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='2 are satisfied for the perfect derived categories of noetherian schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Applying this theorem to such tensor triangulated categories, we obtain the following result: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' (Corollaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='7, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='9, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' (1) Let X be a noetherian quasi-affine scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then there is an isomorphism Spec△(Dpf(X)) ∼= Xred of ringed spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' (2) Let P1 be the projective line over a field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then there is an isomorphism Spec△(Dpf(P1)) ∼= P1 ⊔ �� n∈Z Spec(k) � of ringed spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' (3) Let E be an elliptic curve over an algebraically closed field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then there is an isomorphism Spec△(Dpf(E)) ∼= E ⊔ \uf8eb \uf8ed � (r,d)∈I Er,d \uf8f6 \uf8f8 of ringed spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Here, I := {(r, d) ∈ Z2 | r > 0, gcd(r, d) = 1}, and Er,d is a copy of E for each (r, d) ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Matsui [17](1),(2) and Hirano-Ouchi [12](3) proved that there are homeomorphisms between the underlying spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Therefore, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='3 extends their result to isomorphisms of ringed spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='3(1)(3) shows that reduced quasi-affine schemes and elliptic curves are reconstructed from their perfect derived categories using triangular spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' In Section 2, we give the definitions of the center and the triangular spectra of a triangulated category, which play a central role throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' In Section 3, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='1 and give its ap- plications, including the Bondal-Orlov-Ballard reconstruction theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' In Section 4, we introduce the structure sheaf on the triangular spectrum and prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' We apply this result to the perfect derived categories of noetherian schemes and deduce Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Preliminaries In this section, we recall basic definitions and properties for later use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' We begin with our convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Convention 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' (1) Throughout this paper, we assume that all triangulated cat- egories are essentially small and that all subcategories are full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let T be a triangulated category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' A thick subcategory of T is a subcategory closed under direct summands, shifts, and extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' The set Th(T ) of all thick subcategories forms a lattice with respect to the inclusion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For M ∈ T , denote by ⟨M⟩ the smallest thick subcategory of T containing M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' 4 HIROKI MATSUI (2) For a noetherian scheme X, a complex F of OX-modules is said to be perfect if, for any x ∈ X, there is an open neighborhood U ⊆ X of x such that the restriction F|U is quasi-isomorphic to a bounded complex of locally free sheaves of finite rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Denote by Dpf(X) the derived category of perfect complexes on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' We call it the perfect derived category of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Center of triangulated categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' We recall the definition and basic prop- erties of the center of a triangulated category;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' see [16] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let T be a triangulated category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' (1) The center Z(T ) of T is the set of natural transformations η : idT → idT with η[1] = [1]η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' The composition of natural transformations makes Z(T ) a commutative ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' (2) We say that an element η ∈ Z(T ) is locally nilpotent if ηM is a nilpotent element of the endomorphism ring EndT (M) for each M ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' We shall denote by Z(T )lnil the ideal of Z(T ) consisting of locally nilpotent elements and by Z(T )lred := Z(T )/Z(T )lnil the quotient ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let Φ : T → T ′ be an exact functor between triangulated categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' It seems to be not known whether Φ induces a morphism between Z(T )lred and Z(T ′)lred in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let us give several functoriality results of Z(−)lred under certain functors following [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' We say that an exact functor Φ : T → T ′ is dense if, for any M′ ∈ T ′, there are M ∈ T and N′ ∈ T ′ such that Φ(M) ∼= M′ ⊕ N′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For example, the idempotent completion functor ι : T → T ♮ of T is fully faithful and dense;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' see [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let Φ : T → T ′ be a fully faithful dense exact functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then there is an isomorphism Φ∗ : Z(T ′) ∼ =→ Z(T ), where Φ∗(η)M = Φ−1(ηΦ(M)) for M ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Moreover, this isomorphism induces an isomorphism Φ∗ : Z(T ′)lred ∼ =→ Z(T )lred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' We note that for any object M′ ∈ T ′, there is an object M ∈ T and an isomorphism M′ ⊕ M′[1] ∼= Φ(M);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' see [3, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='2) in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' As Φ : T → T ′ is a fully faithful exact functor, it induces a homomorphism Φ∗ : Z(T ′) → Z(T ) by [16, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='3(1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' First, we prove that this homomorphism is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let η ∈ Z(T ′) with Φ∗(η) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For any M′ ∈ T ′, take an object M ∈ T and an isomorphism M′ ⊕M′[1] ∼= Φ(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then we obtain Φ∗(η)M = Φ−1(ηΦ(M)) ∼= Φ−1(ηM′)⊕Φ−1(ηM′)[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Therefore, Φ∗(η)M = 0 implies Φ−1(ηM′) = 0 and hence ηM′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' This shows that Φ∗ : Z(T ′) → Z(T ) is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' To show that Φ∗ : Z(T ′) → Z(T ) is surjective, fix an element δ ∈ Z(T ) and construct η ∈ Z(T ′) with Φ∗(η) = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For any M′ ∈ T ′, take an object M ∈ T and an isomorphism ϕ : Φ(M) ∼ =−→ M′ ⊕ M′[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then we define the morphism ηM′ : M′ → M′ as the composition M′ ( 1 0) → M′ ⊕ M′[1] ϕ−1 −−→ Φ(M) Φ(δM ) −−−→ Φ(M) ϕ−→ M′ ⊕ M′[1] ( 1 0 ) → M′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' This morphism ηM′ does not depend on the choices of M and ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Indeed, take an object N ∈ T and an isomorphism ψ : Φ(N) ∼ =−→ M′ ⊕ M′[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Since Φ is full, there is TRIANGULAR SPECTRA AND DERIVED CATEGORIES 5 a morphism f : M → N in T such that Φ(f) = ψ−1ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then we have the equalities ϕΦ(δM)ϕ−1 = ψΦ(f)Φ(δM)ϕ−1 = ψΦ(δN)Φ(f)ϕ−1 = ψΦ(δN)ψ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For this reason, ηM′ is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For a morphism g : M′ → N′ in T ′, we prove gηM′ = ηN′g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Take objects M, N ∈ T and isomorphisms ϕ : Φ(M) ∼ =−→ M′ ⊕ M′[1], ψ : Φ(N) ∼ =−→ N′ ⊕ N′[1] in T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Since Φ is full, there is a morphism f : M → N such that Φ(f) = ψ−1 � g 0 0 g[1] � ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then we get the equalities gηM′ = g ( 1 0 ) ϕΦ(δM)ϕ−1 ( 1 0 ) = ( 1 0 ) � g 0 0 g[1] � ϕΦ(δM)ϕ−1 ( 1 0 ) = ( 1 0 ) ψΦ(f)Φ(δM)ϕ−1 ( 1 0 ) = ( 1 0 ) ψΦ(δN)Φ(f)ϕ−1 ( 1 0 ) = ( 1 0 ) ψΦ(δN)ψ−1 � g 0 0 g[1] � ( 1 0 ) = ( 1 0 ) ψΦ(δN)ψ−1 ( 1 0 ) g = ηN′g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' This shows that η : idT ′ → idT ′ is a natural transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Moreover, one can easily see from the definition that η[1] = [1]η holds and hence η ∈ Z(T ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Finally, we will check the equality Φ∗(η) = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For an object M ∈ T , we can take the canonical isomorphism Φ(M ⊕ M[1]) ∼= Φ(M) ⊕ Φ(M)[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Using this isomorphism, we have a commutative diagram Φ(M) ( 1 0) � Φ(δM ) � Φ(M) ⊕ Φ(M)[1] ∼= � Φ(δM ) 0 0 Φ(δM )[1] � � Φ(M ⊕ M[1]) Φ(δM⊕M[1]) � Φ(M) Φ(M) ⊕ Φ(M)[1] ( 1 0 ) � ∼= Φ(M ⊕ M[1]) and this means that ηΦ(M) = Φ(δM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Therefore, Φ∗(η)M = Φ−1(ηΦ(M)) = δM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Thus, we conclude that Φ∗(η) = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' We finish the proof by checking that the first isomorphism induces the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' From the first isomorphism, the induced homomorphism Φ∗ : Z(T ′)lred ։ Z(T )lred is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Pick an element η ∈ Z(T ′) with Φ∗(η) ∈ Z(T )lnil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For any object M ∈ T , there is an integer n > 0 such that Φ−1(ηn Φ(M)) = Φ−1(ηΦ(M))n = Φ∗(η)n M = 0 as Φ∗(η) is locally nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Hence ηn Φ(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Because each object M′ ∈ T ′ is a direct summand of Φ(M) for some M ∈ T , we get η ∈ Z(T ′)lred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' This shows the injectivity of Φ∗ : Z(T ′)lred ։ Z(T )lred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' ■ For thick subcategories U ⊆ V of T , there is a unique exact functor QV/U : T /U → T /V such that QV/U ◦ QU = QV, where QU : T → T /U and QV : T → T /V are the canonical functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let U ⊆ V be thick subcategories of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then the canonical functor QV/U : T /U → T /V induces a ring homomorphism (QV/U)∗ : Z(T /U) → Z(T /V) 6 HIROKI MATSUI where (QV/U)∗(η)QV(M) = QV/U(ηQU(M)) for M ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Moreover, this homomorphism induces a homomorphism (QV/U)∗ : Z(T /U)lred → Z(T /V)lred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' It follows from [23, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='1] that V/U is a thick subcategory of T /U and that QV/U induces a triangle equivalence Φ : (T /U)/(V/U) ∼ =−→ T /V such that Φ ◦ Q = QV/U, where Q : T /U → (T /U)/(V/U) is the canonical functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='3 and [16, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='3(2)], we obtain the homomorphism (QV/U)∗ : Z(T /U) Q∗ −→ Z((T /U)/(V/U)) (Φ∗)−1 −−−−→ ∼ = Z(T /V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' The second statement is clear from the description (QV/U)∗(η)QV(M) = QV/U(ηQU(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' ■ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Spectra of triangulated categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' In this subsection, let us recall the two kinds of spectra introduced by Balmer [3] and Matsui [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let T be a triangulated category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' We set Z(E) := {X ∈ Th(T ) | X ∩ E = ∅} for a subcategory E of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then we can easily see the following properties: (i) Z(T ) = ∅ and Z(∅) = Th(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' (ii) � i∈I Z(Ei) = Z(� i∈I Ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' (iii) Z(E) ∪ Z(E′) = Z(E ⊕ E′), where E ⊕ E′ := {M ⊕ M′ | M ∈ E, M′ ∈ E′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Therefore, we can define the topology on Th(T ) whose closed subsets are of the form Z(E) for some E ⊆ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' A tensor triangulated category is a triple (T , ⊗, 1) consisting of a triangulated cat- egory T together with a symmetric monoidal structure (⊗, 1) such that the bifunctor ⊗ : T × T → T is exact in each variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' ([3, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let (T , ⊗, 1) be a tensor triangulated category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' A proper thick subcategory P ⊊ T is said to be a prime thick ideal if it satisfies the following conditions: (ideal) M ⊗ N ∈ P holds for any M ∈ T and N ∈ P, (prime) M ⊗ N ∈ P implies M ∈ P or N ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Denote by Spec⊗(T ) the set of prime thick ideals of T together with the induced topology of Th(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' We call Spec⊗(T ) the Balmer spectrum or the tensor-triangular spectrum of (T , ⊗, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Balmer [3] also defined the structure sheaf on Spec⊗(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For an open subset U ⊆ Spec⊗(T ), set T U := � P∈U P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' We define the tensor triangulated category T (U) by T (U) := � T /T U�♮ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Denote by 1U the image of 1 under the canonical functor T → T (U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' TRIANGULAR SPECTRA AND DERIVED CATEGORIES 7 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' ([3, Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' We denote by O⊗,T the sheafification of the presheaf Op ⊗,T of commutative rings given by the assignment U �→ Op ⊗,T (U) := EndT (U)(1U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' We simply write Spec⊗(T ) for the ringed space (Spec⊗(T ), O⊗,T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Using the Balmer spectrum, Balmer [3] proved that any noetherian scheme X could be reconstructed from the tensor triangulated category (Dpf(X), ⊗L OX, OX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' ([3, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='3(a)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For a noetherian scheme X, there is an isomorphism SX : X ∼ =−→ Spec⊗(Dpf(X)) of ringed spaces, where the map of underlying topological spaces is given by x �→ SX(x) := {F ∈ Dpf(X) | Fx ∼= 0 in Dpf(OX,x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Next, let us recall the triangular spectrum of a triangulated category of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' ([17, Definitions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let T be a triangulated category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' A proper thick subcategory P ⊊ T is called a prime thick subcategory if the partially ordered set {X ∈ Th(T ) | P ⊊ X } has the smallest element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Denote by Spec△(T ) the set of prime thick subcategories of T together with the induced topology of Th(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' We call Spec△(T ) the triangular spectrum of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' The above definition of a prime thick subcategory differs from that in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' There, we call P a prime thick subcategory if {X ∈ Th(T ) | P ⊊ X } has a unique minimal element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' The correct definition is the one in this paper, and all the proofs in [17] are done using the present definition (“unique minimal element” in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='15 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='7 should be changed to “smallest element”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Also, there are no problems in the results in [12] if we adopt the definition in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let Φ : T → T ′ be an exact functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Since Φ−1(X ) := {M ∈ T | Φ(M) ∈ T ′} is a thick subcategory of T for each thick subcategory X ⊆ T ′, we have an order- preserving map Φ∗ : Th(T ′) → Th(T ), X �→ Φ−1(X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' This map restricts to continuous maps between triangular spectra for fully faithful dense exact functors and quotient functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' ([17, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let Φ : T → T ′ be a fully faithful dense exact functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then the map Φ∗ : Th(T ′) → Th(T ) restricts to a homeomorphism Φ∗ : Spec△(T ′) ∼ =→ Spec△(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' ([17, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let T be a triangulated category and U be its thick subcategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Denote by Q : T → T /K the canonical functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then the map Q∗ : Th(T /K) → Th(T ) restricts to an immersion Q∗ : Spec△(T /K) ֒→ Spec△(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' of topological spaces whose image is {P ∈ Spec△(T ) | K ⊆ P}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' 8 HIROKI MATSUI The following theorem is one of the main results in [17], which is some sort of a generalization of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let X be a noetherian scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' (1) Let P be a thick ideal of Dpf(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then P is a prime thick subcategory if and only if P is a prime thick ideal if and only if P = SX(x) for some x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' (2) We have an immersion SX : X ֒→ Spec△(Dpf(X)), x �→ SX(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' of topological spaces whose image is Spec⊗(Dpf(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' We will see in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='7 that the above immersion SX : X ֒→ Spec△(Dpf(X)) follows from more general result Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' From the definition, it is quite difficult to determine the topological space Spec△(Dpf(X)) for a given noetherian scheme X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For special cases, triangular spec- tra are determined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' (1) ([17, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='17(1)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' If X is a quasi-affine noetherian scheme, then there is a homeomorphism Spec△(Dpf(X)) ∼= X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' (2) ([17, Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let P1 be the projective line over a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then there is a homeomorphism Spec△(Dpf(P1)) ∼= P1 ⊔ Z, where Z is considered as the discrete topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' (3) ([12, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let E be an elliptic curve over a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then there is a homeomorphism Spec△(Dpf(E)) ∼= E ⊔ \uf8eb \uf8ed � (r,d)∈I Er,d \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Here, I := {(r, d) ∈ Z2 | r > 0, gcd(r, d) = 1} and Er,d is a copy of E for each (r, d) ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Reconstruction of schemes In this section, we discuss the reconstruction of a noetherian scheme X from the triangulated category Dpf(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Reconstruction of underlying topological spaces has been discussed in [12, 17] in terms of the triangular spectrum of Dpf(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' To reconstruct the structure sheaf, centers of triangulated categories play a crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' We recall that a tensor triangulated category (T , ⊗, 1) is rigid if it is closed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=', the exact functor M ⊗ − : T → T has a right adjoint [M, −] : T → T for each M ∈ T and such that every object M is strongly dualizable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=', the canonical map [M, 1] ⊗ N → [M, N] is an isomorphism for each N ∈ T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' see [13] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' If T is rigid, then so is T (U) for any open subset U ⊆ Spec⊗(T ) by [4, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' TRIANGULAR SPECTRA AND DERIVED CATEGORIES 9 We say that the tensor triangulated category (T , ⊗, 1) is monogenic if T is gener- ated by 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=', T = ⟨1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' We say that T is locally monogenic if, for any prime thick ideal P ∈ Spec⊗(T ), there is a quasi-compact open neighborhood U ⊆ Spec⊗(T ) of P such that T (U) is monogenic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' We begin with stating the following general result for specific tensor triangulated categories whose proof is taken from [19, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let (T , ⊗, 1) be an idempotent complete, rigid, locally monogenic tensor triangulated category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then the evaluation at 1 induces an isomorphism Z(T )lred ∼= EndT (1)red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' As Spec⊗(T ) is a spectral space, there are quasi-compact open covering U1, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' , Un of Spec⊗(T ) such that T (Ui) = ⟨1Ui⟩ for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let α : Z(T ) → EndT (1), η �→ η1 be the evaluation at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Since α has a right inverse EndT (1) → Z(T ), φ �→ φ ⊗ (−), the homomorphism α is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' There- fore, it suffices to show that the induced homomorphism α : Z(T )lred → EndT (1)red is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' To this end, let us prove η ∈ Z(T )lnil for each η ∈ Z(T ) with α(η) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=', η1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' We proceed by induction on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' First assume n = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=', T = ⟨1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' We prove that ηM is nilpotent for any M ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' We consider the subcategory X ⊆ T consisting of objects M ∈ T with ηM nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then X contains 1 as η1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' In addition, X is a thick subcategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Indeed, it is clear that X is closed under direct summands and shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Take an exact triangle L f−→ M g−→ N → L[1] in T with L, N ∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then there is an integer l ≥ 1 such that ηl L = 0 and ηl N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' From the naturality of ηl, the equality gηl M = ηl Ng = 0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Thus, ηl M factors as ηl M : M a−→ L f−→ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Again using the naturality of ηl, we get η2l M = ηl Mηl M = ηl Mfa = fηl La = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' As a result, M ∈ X follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Therefore, X is thick and so X = ⟨1⟩ = T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' This means that η is locally nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Next, assume that n > 1 and set V = U2 ∪ U3 ∪ · · ·Un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Assume further that the evaluations at the unit objects induce isomorphisms α : Z(T (U1))lred ∼ =−→ EndT (U1)(1U1)red, α : Z(T (V ))lred ∼ =−→ EndT (V )(1V )red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Here, we note that there is a homeomorphism f : Spec⊗(T (V )) ∼ =−→ V such that T (V )(f −1(Ui)) ∼= T (Ui) = ⟨1Ui⟩ for i = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' , n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' see [7, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='11] and [6, Constructions 24 and 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Hence we can apply the induction hypothesis to T (V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' One can easily verify that the canonical functors Q : T → T /T U1 and ι : T /T U1 → T (U1) induce a commutative diagram Z(T )lred Q∗ � α � Z(T /T U1)lred α � Z(T (U1))lred α ∼ = � ι∗ ∼ = � EndT (1)red Q � EndT /T U1(Q(1))red ι ∼ = � EndT (U1)(1U1)red, where the evaluations at the unit objects induce the vertical arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' From this diagram, η1 = 0 yields Q∗(η) ∈ Z(T /T U1)lnil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For an object M ∈ T , there is an 10 HIROKI MATSUI integer l ≥ 1 such that Q(ηl M) = Q∗(η)l Q(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Accordingly, ηl M factors some N ∈ T U1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then η(d+1)l M factors ηdl N for any integer d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For the canonical functors Q′ : T → T /T V and ι′ : T /T V → T (V ), the same argument as above shows that Q′ ∗(η) ∈ Z(T /T V )lnil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Here we use the isomorphism α : Z(T (V ))lred ∼ =−→ EndT (V )(1V )red which is our induction hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' In particular, Q′(ηdl N) = Q′ ∗(η)dl Q′(N) = 0 for some d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' It follows from [7, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='1] that the functor Q′ restricts to a fully faithful functor Q′ : T U1 → (T /T V )U1 and hence Q′(ηdl N) = 0 implies ηdl N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then η(d+1)l M = 0 holds as η(d+1)l M factors ηdl N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' As a result, we conclude that η ∈ Z(T )lnil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' ■ Let X be a noetherian scheme and U ⊆ X be an open subset with Z := X \\ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then [2, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='13] shows that the restriction functor (−)|U: Dpf(X) → Dpf(U) induces a triangle equivalence � Dpf(X)/Dpf Z (X) �♮ ∼= Dpf(U) of tensor triangulated categories, where Dpf Z (X) = {F ∈ Dpf(X) | F|U∼= 0} = � x∈U SX(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Therefore, the left-hand side is Dpf(X)(SX(U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Applying Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='1 to T = Dpf(X), the following result is recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' ([2, Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='1] and [19, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For a noetherian scheme X, there is an isomorphism Z(Dpf(X))lred ∼= Γ(X, OX)red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Dpf(X) is idempotent complete because it is realized as the full subcategory of compact objects of Dqc(Mod OX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Also, it is rigid by [4, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For any affine scheme U, it holds that Dpf(U) = ⟨OU⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' It follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='7 and [2, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='13] that the tensor triangulated category Dpf(X) is locally monogenic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Thus, we get isomorphisms Z(Dpf(X))lred ∼= EndDpf(X)(OX)red ∼= Γ(X, OX)red of commutative rings from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' ■ Now we state and prove the first main result in this paper about the reconstruction of a noetherian scheme from the perfect derived category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let X1 and X2 be noetherian schemes and Φ : Dpf(X1) ∼ =−→ Dpf(X2) be a triangle equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Assume that there is a line bundle Li on Xi for i = 1, 2 satisfying the following conditions: (i) Li is ample or anti-ample for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' (ii) There is an isomorphism Φ(F ⊗L OX1 L1) ∼= Φ(F) ⊗L OX2 L2 for any F ∈ Dpf(X1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then (X1)red and (X2)red are isomorphic as schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' TRIANGULAR SPECTRA AND DERIVED CATEGORIES 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For i = 1, 2, denote by SpecLi △ (Dpf(Xi)) the subset of Spec△(Dpf(Xi)) con- sisting of prime thick subcategories P with P ⊗L OXi Li := {F ⊗L OXi Li | F ∈ P} = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then the homeomorphism Φ∗ : Spec△(Dpf(X2)) ∼ =−→ Spec△(Dpf(X1)) restricts to the homeomorphism Φ∗ : SpecL2 △ (Dpf(X2)) ∼ =−→ SpecL1 △ (Dpf(X1)) by (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' On the other hand, as Li is ample or anti-ample by (i), {L⊗n i | n ∈ Z} generates Dpf(Xi), and hence Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='12(1) implies SpecLi △ (Dpf(X1)) = Spec⊗(Dpf(Xi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' From Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='7, we obtain a homeomorphism SXi : Xi ∼ =−→ Spec⊗(Dpf(X1)) = SpecLi △ (Dpf(Xi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Thus, we get a homeomorphism f : X2 SX2 −−→ SpecL2 △ (Dpf(X2)) Φ∗ −→ SpecL1 △ (Dpf(X1)) S−1 X1 −−→ X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' The construction of this map yields Φ(SX1(f(x2))) = SX2(x2) for any x2 ∈ X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For an open subset U ⊆ X2 with Z = X2 \\ U, one has Φ(Dpf f (Z )(X1)) = Φ \uf8eb \uf8ed � x1∈f(Z) SX1(x1) \uf8f6 \uf8f8 = � x1∈f(Z) Φ (SX1(x1)) = � x2∈Z SX2(x2) = Dpf Z (X2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Therefore, the triangle equivalence Φ : Dpf(X1) ∼ =−→ Dpf(X2) induces a triangle equiv- alence Φ : Dpf(X1)/Dpf f (Z )(X1) ∼ =−→ Dpf(X2)/Dpf Z (X2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Taking idempotent completion and Z(−)lred, we obtain an isomorphism Γ(f(U), OX1)red ∼= Γ(U, OX2)red by [2, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='13] and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' As we can easily see that this isomorphism is compatible with the restriction of open subsets, we get an isomorphism (X2)red ∼= (X1)red of schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' ■ This result has several applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' The first one is the following reconstruction theorem which is well-known at least for the affine case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let X1 and X2 be noetherian quasi-affine schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' If there is a tri- angle equivalence Φ : Dpf(X1) ∼ =−→ Dpf(X2), then (X1)red and (X2)red are isomorphic as schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Take Li to be OXi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then OXi is an ample line bundle as Xi is quasi-affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Moreover, the isomorphism Φ(F ⊗L OX1 OX1) ∼= Φ(F) ⊗L OX2 OX2 obviously holds for each F ∈ Dpf(X1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' hence, the result follows by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' ■ Another application of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='3 is to recover the famous result by Bondal- Orlov [9] and Ballard [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' ([1, 9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let Xi be a Gorenstein projective scheme over a field k with ample or anti-ample canonical bundle ωXi for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' If there is a k-linear triangle 12 HIROKI MATSUI equivalence Φ : Dpf(X1) ∼ =−→ Dpf(X2), then (X1)red and (X2)red are isomorphic as schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Take Li to be ωXi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then the assumption (i) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Recall that a Serre functor SXi : Dpf(Xi) → Dpf(Xi) on Dpf(Xi) is a triangle equivalence such that there is a natural isomorphism HomDpf(Xi)(F, G) ∼= HomDpf(Xi)(G, SXi(F))∗ for any F, G ∈ Dpf(Xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Here (−)∗ stands for the k-dual functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' It follows from [1, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='6] that SXi ∼= (−) ⊗L OXi ωXi[− dim Xi] and from [14, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='30] that SX2 ◦ Φ ∼= Φ ◦ SX1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Therefore, the assumption (ii) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Applying Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='3, we get an isomorphism between (X1)red and (X2)red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' ■ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' (1) The original statement by Bondal-Orlov [9] and Ballard [1] as- sumes that either ωX1 or ωX2 is ample or anti-ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Although Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='5 requires both of them to be ample or anti-ample, our argument is purely alge- braic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' (2) There is a generalization of the Bondal-Orlov-Ballard reconstruction theorem to the relative case by Sancho de Salas and Sancho de Salas [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='3 also recovers their result by the same argument as in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Triangular spectra as ringed spaces One of the key ingredients of proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='3 is the isomorphisms Z \uf8eb \uf8ed � Dpf(X)/ � x∈U SX(x) �♮\uf8f6 \uf8f8 lred ∼= Z � Dpf(U) � lred ∼= Γ(U, OX)red for an open subset U ⊆ X, which is a consequence of [2, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='13] and Corol- lary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Motivated by these isomorphisms, we define the structure sheaf on the triangular spectrum for a triangulated category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let T be a triangulated category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For an open subset U ⊆ Spec△(T ), we set T U := � P∈U P, T (U) := � T /T U�♮ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' By composing the quotient functor QU : T → T /T U and the idempotent completion functor ιU : T /T U → T (U), we have a natural functor resU : T → T (U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Thanks to Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='4, the functor resU induces a homomorphism (resU)∗ : Z(T )lred (QU)∗ −−−→ Z(T /T U)lred ((ιU )∗)−1 −−−−−→ ∼ = Z(T (U))lred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let U ⊆ V ⊆ Spec△(T ) be open subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then the universal properties of the Verdier quotient and the idempotent completion shows that the inclusion T V ⊆ T U induces a unique exact functor resV,U : T (V ) → T (U) TRIANGULAR SPECTRA AND DERIVED CATEGORIES 13 such that resV,U ◦resV = resU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Again using Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='4, this functor induces a homomorphism (resV,U)∗ : Z(T (V ))lred → Z(T (U))lred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Furthermore, we have the equality (resW,V )∗ ◦ (resV,U)∗ = (resW,U)∗ for three open subsets U ⊆ V ⊆ W ⊆ Spec△(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let T be a triangulated category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' We define the sheaf O△,T of commutative rings on Spec△(T ) by the sheafification of the presheaf Op △,T : Op △,T (U) := Z(T (U))lred, ρV,U := (resV,U)∗ : Op △,T (V ) → Op △,T (U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Later, we consider the spectrum Spec△(T ) as the ringed space (Spec△(T ), O△,T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Recall that a morphism (f, f ♭) : (X, OX) → (Y, OY ) of ringed spaces consists of a continuous map f : X → Y and a morphism f ♭ : f −1OY → OX of sheaves of commutative rings on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' We say that the morphism (f, f ♭) is an open immersion of ringed spaces if f : X → Y is an open immersion of topological spaces and f ♭ : f −1OY → OX is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' We remark that for a continuous map f : X → Spec△(T ), the pullback sheaf f −1O△,T is isomorphic to the sheafification of the presheaf f pOp △,T : U �→ lim −→ f(U)⊆V Op △,T (V ) of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='4 show that the continuous maps in Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='10 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='11 can be extended to morphisms of ringed spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let Φ : T → T ′ be a fully faithful dense exact functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then we have an isomorphism Φ∗ : Spec△(T ′) ∼ =−→ Spec△(T ) of ringed spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' This follows from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='3 and Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' ■ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let U be a thick subcategory of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then the quotient functor Φ : T → T /U induces a morphism Φ∗ : Spec△(T /K) → Spec△(T ) of ringed spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Moreover, if the image of Φ∗ is an open subset of Spec△(T ), then the above morphism is an open immersion of ringed spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' We construct a morphism (Φ∗)♭ : (Φ∗)−1O△,T → O△,T /K of sheaves of com- mutative rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Take open subsets U ⊆ Spec△(T /K) and V ⊆ Spec△(T ) with Φ∗(U) ⊆ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Setting �U := Φ∗(U) = {P ∈ Spec△(T ) | P/K ∈ U}, one sees (T /K)U = � P∈ �U P/K = T �U/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then the inclusion T V ⊆ T �U induces a triangle functor T /T V → T /T �U ∼= (T /K)/(T �U/K) = (T /K)/(T /K)U, where the triangle equivalence is by [23, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='1(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Applying Z((−)♮), we get a homomorphism Op △,T (V ) → Op △,T /K(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Moreover, let U ⊆ U′ ⊆ Spec△(T /K) 14 HIROKI MATSUI and V ⊆ V ′ ⊆ Spec△(T ) be other open subsets with � U′ := Φ∗(U′) ⊆ V ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then the inclusions T V � � � T �U T V ′ � � � �� � T � U′ �� � induce a commutative diagram T /T V � T /T �U ∼= (T /K)/(T /T �U) T /T V ′ � � T /T � U′ � ∼= (T /K)/(T /T � U′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' � Applying Z((−)♮)lred, we obtain a commutative diagram Op △,T (V ) � Op △,T /K(U) Op △,T (V ′) � � Op △,T /K(U′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' � Therefore, the homomorphism Op △,T (V ) → Op △,T /K(U) defines a morphism (Φ∗)pOp △,T → Op △,T /K of presheaves of commutative rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Taking sheafifications, we obtain a morphism (Φ∗)♭ : (Φ∗)−1O△,T → O△,T /K of sheaves of commutative rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Next, assume Φ∗(Spec△(T /K)) ⊆ Spec△(T ) is an open subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then Φ∗ : Spec△(T /K) → Spec△(T ) is an open immersion of topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For any open subset U ⊆ Spec△(T /K), the map (Φ∗)pOp △,T /K(U) → Op △,T (U) is induced from the triangle equivalence T /T �U ∼= (T /K)/(T �U/K) = (T /K)/(T /K)U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' As a result, the map (Φ∗)pOp △,T /K → Op △,T is an isomorphism and hence so is (Φ∗)♭ : (Φ∗)−1O△,T /K → O△,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' ■ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For an object M ∈ T , the quotient functor Φ : T → T /⟨M⟩ induces an open immersion Φ∗ : Spec△(T /⟨M⟩) ֒→ Spec△(T ) of ringed spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' The image of Φ∗ is {P ∈ Spec△(T ) | M ∈ T } = Spec△(T ) \\ Z({M}), which is an open subset of Spec△(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' The result follows from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' ■ The main theorem in this section compares the Balmer spectrum and the tri- angular spectrum for an idempotent complete, rigid, and locally monogenic tensor triangulated category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' We treat the monogenic case first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' TRIANGULAR SPECTRA AND DERIVED CATEGORIES 15 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let T be an idempotent complete, rigid, monogenic tensor triangu- lated category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Assume further that Spec⊗(T ) is noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then the equality Spec△(T ) = Spec⊗(T ) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Since T = ⟨1⟩, every thick subcategory is a thick ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Therefore, by [3, Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='3], the thick subcategories and the radical thick ideals coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let P ∈ Spec△(T ) and take the smallest element X in {X ∈ Th(T ) | P ⊊ X }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Since P is a radical thick ideal by the above argument, it is the intersection of all prime thick ideals Q containing P by [3, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' If P is not a prime thick ideal, then such Q satisfies P ⊊ Q and hence X ⊆ Q by the assumption on X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Therefore, P ⊊ X ⊆ � Q∈Spec⊗(T ),P⊆Q Q = P leads a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Hence we conclude that the inclusion Spec△(T ) ⊆ Spec⊗(T ) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Conversely, take P ∈ Spec⊗(T ) and prove P ∈ Spec△(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' It follows from [3, The- orem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='10] that there is an order-preserving bijection between the thick subcategories of T and the specialization-closed subsets of Spec⊗(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' The noetherian assumption is used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Under this bijection, P corresponds to the specialization-closed subset W := {Q ∈ Spec⊗(T ) | P ̸⊆ Q} = {Q ∈ Spec⊗(T ) | P ̸∈ {Q}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Here we use [3, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Therefore, it suffices to show that there is the smallest specialization-closed subset T with W ⊊ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Set T := W ∪ {P} and prove that this is the specialization-closed subset that we need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For an element Q ∈ {P} \\ {P}, Q belongs to W as P ̸⊆ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Thus T = W ∪ {P} holds and this is specialization-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For a specialization-closed subset W ′ ⊆ Spec⊗(T ) with W ⊊ W ′, we will check T ⊆ W ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Since W ⊊ W ′, there is an element Q ∈ W ′ such that Q ̸∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then Q ̸∈ W implies P ∈ {Q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' As W ′ is specialization-closed, one has P ∈ {Q} ⊆ W ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' This shows that the inclusion T ⊆ W ′ and hence T is the smallest specialization-closed subset with W ⊊ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' ■ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let (T , ⊗, 1) be an idempotent complete, rigid, locally monogenic tensor triangulated category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Assume further that Spec⊗(T ) is noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then there is a morphism i : Spec⊗(T )red → Spec△(T ) of ringed spaces satisfying the following conditions: (1) The map of underlying topological spaces is the inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' In particular, every prime thick ideal is a prime thick subcategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' (2) i is an open immersion of ringed spaces whenever Spec⊗(T ) is an open subset of Spec△(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' (3) i is an isomorphism of ringed spaces if T is monogenic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' First, we prove (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let P ∈ Spec⊗(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Take a quasi-compact open subset P ∈ U ⊆ Spec⊗(T ) with T (U) = ⟨1U⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then T (U) is also idempotent complete, rigid, and locally monogenic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' On the other hand, it follows from [7, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='11] that the canonical functor resU : T → T (U) induces a homeomorphism (resU)∗ : Spec⊗(T (U)) ∼ =−→ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' In particular, Spec⊗(T (U)) is noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Applying Lemma 16 HIROKI MATSUI 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='5 yields that Spec⊗(T (U)) = Spec⊗(T (U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then we obtain the inclusion U ⊆ Spec△(T (U)) as the composition U ((resU)∗)−1 −−−−−−→ Spec⊗(T (U)) = Spec△(T (U)) (resU)∗ −−−−→ Spec△(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Therefore, P ∈ U ⊆ Spec△(T (U)) and hence we get Spec⊗(T ) ⊆ Spec△(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Next, let us construct a morphism i−1O△,T → (O⊗,T )red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For open subsets U ⊆ Spec⊗(T ) and V ⊆ Spec△(T ) with U ⊆ V , the inclusion T V ⊆ T U induces a homomorphism Op △,T (V ) = Z (T (V ))lred → Z (T (U))lred ∼= EndT (U)(1U)red = Op ⊗,T (U)red, where the isomorphism is proved in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Moreover, for other open subsets U ⊆ U′ ⊆ Spec⊗(T ) and V ⊆ V ′ ⊆ Spec△(T ) with U′ ⊆ V ′, we get the commutative diagram T /T V � T /T U T /T V ′ � � T /T U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' � Applying Z((−)♮)lred yields a commutative diagram Op △,T (V ) Z(T (V ))lred � Z(T (U))lred ∼= Op ⊗,T (U)red Op △,T (V ′) � Z(T (V ′))lred � � Z(T (U′))lred � ∼= Op ⊗,T (U′)red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' � Therefore, Op △,T (V ) → Op ⊗,T (U)red defines a morphism ipOp △,T → (Op ⊗,T )red of presheaves of commutative rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Taking sheafifications, we obtain a morphism i♭ : i−1O△,T → (O⊗,T )red of sheaves of commutative rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Assume Spec⊗(T ) is an open subset of Spec△(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then for any open subset U ⊆ Spec⊗(T ), the homomorphism ipOp △,T (U) → Op ⊗,T (U) is given by the isomorphism Z(T (U))lred ∼= EndT (U)(1U)red in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Thus, ipOp △,T (U) → Op ⊗,T (U) is an isomorphism and hence so is i−1O△,T (U) → O⊗,T (U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' This means that i : Spec⊗(T ) → Spec△(T ) is an open immersion of ringed spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' In particular, i : Spec⊗(T ) → Spec△(T ) is an isomorphism of ringed spaces if T is monogenic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' ■ As Dpf(X) is an idempotent complete, rigid, locally monogenic tensor triangu- lated category, we get the following immediate consequence of the combination of Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='7 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For a noetherian scheme X, there is a morphism of ringed spaces SX : Xred → Spec△(Dpf(X)), x �→ SX(x), which is an open immersion of ringed spaces if SX(X) is open in Spec△(Dpf(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' In particular, SX : Xred → Spec△(Dpf(X)) is an isomorphism of ringed spaces if X is quasi-affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' TRIANGULAR SPECTRA AND DERIVED CATEGORIES 17 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Since the ringed space Spec△(Dpf(X)) is completely determined by the triangulated category structure on Dpf(X), Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='7 implies Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Moreover, using Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='7, we determine Spec△(Dpf(X)) for X which appeared in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let P1 be the projective line over a field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then there is an iso- morphism Spec△(Dpf(P1)) ∼= P1 ⊔ �� n∈Z Spec(k) � of ringed spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For any integer n ∈ Z, it follows from [14, Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='29] that we get triangle equivalences Dpf(P1)/⟨OP1(n)⟩ ∼= ⟨OP1(n + 1)⟩ ∼= Dpf(Spec(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' It follows from Corollaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='7 that there is an open immersion fn : Spec(k) SSpec(k) −−−−→ ∼ = Spec△(Dpf(Spec(k))) ∼= Spec△(Dpf(P1)/⟨OP1(n)⟩) ֒→ Spec△(Dpf(P1)) of ringed spaces whose image is ⟨OP1(n)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Moreover, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='7 and [12, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='7] show that there is an open immersion g := SP1 : P1 ֒→ Spec△(Dpf(P1)) of ringed spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' As it is proved in [17, Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='10], Spec△(Dpf(P1)) is the disjoint union of the images of fn (n ∈ Z) and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Therefore, fn (n ∈ Z) and g induce isomorphism P1 ⊔ �� n∈Z Spec(k) � ∼= Spec△(Dpf(P1)) of ringed spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' ■ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let E be an elliptic curve over an algebraically closed field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then there is an isomorphism Spec△(Dpf(E)) ∼= E ⊔ \uf8eb \uf8ed � (r,d)∈I Er,d \uf8f6 \uf8f8 of ringed spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Here, I := {(r, d) ∈ Z2 | r > 0, gcd(r, d) = 1} and Er,d is a copy of E for each (r, d) ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' For an element (r, d) ∈ I, M(r, d) denotes the moduli space of µ-semistable sheaves with Chern character (r, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' It follows from [22, Theorem 1] that M(r, d) ∼= Er,d, where Er,d is a copy of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Moreover, [11, Proposition 3] shows that there is a triangle equivalence Φr,d : Dpf(E) ∼ =−→ Dpf(M(r, d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Then we get open immersions fr,d : Er,d ∼= M(r, d) SM(r,d) ֒→ Spec△(Dpf(M(r, d))) Φ∗ r,d −−→ ∼ = Spec△(Dpf(E)), 18 HIROKI MATSUI g := SE : E ֒→ Spec△(Dpf(E)) of ringed spaces by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='7 and [12, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' It is shown in [12, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='11] that Spec△(Dpf(E)) is the disjoint union of open subsets fr,d(Er,d) ((r, d) ∈ I) and g(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Hence fr,d ((r, d) ∈ I) and g induce an isomorphism E ⊔ \uf8eb \uf8ed � (r,d)∈I Er,d \uf8f6 \uf8f8 ∼ =−→ Spec△(Dpf(E)) of ringed spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' ■ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Let E and E′ be elliptic curves over an algebraically closed field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' If there is a triangle equivalence Dpf(E) ∼= Dpf(E′), then Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='10 shows that there is an isomorphism ⊔n∈ZE ∼= ⊔n∈ZE′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' As their connected components, this implies that E and E′ are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' This fact is well-known to experts and can be found in [14, Pages 134 and 135] for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Our argument is completely different from the known argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Ballard, Derived categories of sheaves on singular schemes with an application to 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+page_content=' Verdier, Des cat´egories d´eriv´ees des cat´egories ab´eliennes, Ast´erisque, 239 (1996), xii+253pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content=' Department of Mathematical Sciences, Faculty of Science and Technology, Tokushima University, 2-1 Minamijyousanjima-cho, Tokushima 770-8506, Japan Email address: hmatsui@tokushima-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} +page_content='jp' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfbASE/content/2301.03168v1.pdf'} diff --git a/WdE0T4oBgHgl3EQf3ALu/content/2301.02721v1.pdf b/WdE0T4oBgHgl3EQf3ALu/content/2301.02721v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..64aaf7481508e259c968d5e572c321dbf8af0d61 --- /dev/null +++ b/WdE0T4oBgHgl3EQf3ALu/content/2301.02721v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3377a31a21fd8cc397413f9f0d41624b982fb6f203a13a12015e08b8eef64b8e +size 39475128 diff --git a/WtE0T4oBgHgl3EQf3QI4/content/tmp_files/2301.02722v1.pdf.txt b/WtE0T4oBgHgl3EQf3QI4/content/tmp_files/2301.02722v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6566720c2c58c5467756024353e0a1fe095dc1b3 --- /dev/null +++ b/WtE0T4oBgHgl3EQf3QI4/content/tmp_files/2301.02722v1.pdf.txt @@ -0,0 +1,2504 @@ +arXiv:2301.02722v1 [gr-qc] 6 Jan 2023 +Covariant definition of Double Null Data +and geometric uniqueness of the +characteristic initial value problem +Marc Mars∗ and Gabriel S´anchez-P´erez† +Departamento de F´ısica Fundamental, Universidad de Salamanca +Plaza de la Merced s/n, 37008 Salamanca, Spain +January 10, 2023 +Abstract +The characteristic Cauchy problem of the Einstein field equations has been +recently addressed from a completely abstract viewpoint by means of hypersurface +data and, in particular, via the notion of double null data. However, this definition +was given in a partially gauge-fixed form. In this paper we generalize the notion +of double null data in a fully diffeomorphism and gauge covariant way, and show +that the definition is complete by proving that no extra conditions are needed to +embed the double null data in some spacetime. The second aim of the paper is +to show that the characteristic Cauchy problem satisfies a geometric uniqueness +property. Specifically, we introduce a natural notion of isometry at the abstract +level such that two double null data that are isometric in this sense give rise to +isometric spacetimes. +1 +Introduction +The aim of this paper is two-fold. Firstly, we prove a geometric uniqueness result of +the Characteristic Cauchy problem in General Relativity. A prerequisite for a result +of this type is to have an abstract notion of the initial data completely detached from +the spacetime one wishes to construct. In the standard Cauchy problem this consists of +a Riemannian manifold endowed with a symmetric (0,2)-tensor satisfying the vacuum +constraint equations. In the null case, not such abstract formulation was known until +recently. In [13] we have developed a fully detached notion of initial data for the char- +acteristic problem. The key notions to achieve this are the hypersurface data formalism, +the concept of double null data and the so-called constraint tensor. However, the defini- +tion of double null data in [13] was given in a (partially) gauge-fixed form. The second +objective of this paper is to address this problem and give a fully gauge-covariant notion +of double null data. +The hypersurface data formalism was developed in [11, 10] to study general hypersur- +faces at a purely abstract level, i.e., without reference to any ambient space. It has been +employed recently in the problem of matching of spacetimes across null hypersurfaces +[9] and to study the characteristic problem in General Relativity [13]. A hypersurface +∗marc@usal.es +†gasape21@usal.es +1 + +data set D consists of an abstract manifold H and a set of tensors defined on H, with +which one can reconstruct the full ambient metric on the hypersurface and the transverse +derivative of its tangential components whenever the data happens to be embedded. Hy- +persurface data has an internal gauge structure associated to the freedom in the choice +of an everywhere transverse vector (so-called rigging) to the hypersurface in the embed- +ded case. In [13] we have particularized the definition of hypersurface data to the null +case, so that it describes an abstract null hypersurface. It turns out that that when +the data is embedded, the pullback of the ambient Ricci tensor into the null hypersur- +face can be written completely in terms of the abstract data. This allowed us to define +and study the characteristic problem for the Einstein equations in a completely abstract +way. The data corresponds to two null hypersurfaces intersecting transversely in a space- +time. When the full metric is prescribed in two such null hypersurfaces, Randall [15] +has shown that the reduced Einstein equations are well-posed. Moreover, if the metric +components are prescribed suitably, the ambient spacetime is not only a solution of the +reduced equations, but also a solution of the vacuum Einstein equations. This spacetime +exists in a neighbourhood of the intersection and the result is valid on any dimension +and for all topologies of the intersection submanifold. When the intersecting surface is +a 2-sphere, Luk shows in [7] that the spacetime can be extended to a neighbourhood of +the two initial null hypersurfaces. In dimension four the existence of the solution in a +full neighbourhood of the initial data hypersurfaces has been proved in [1] for a large +class of symmetric hyperbolic systems (including Einstein equations) irrespectively of +the topology of the intersection. Other approaches to the characteristic problem can be +found in [2, 4, 5, 6, 14]. In [3] this Cauchy problem is studied on the future null cone of +a point. +As already mentioned, in [13] we approached the characteristic initial value problem in a +fully abstract way by means of the hypersurface data formalism, and in particular with a +new geometric object called double null data. Our formulation detaches the initial data +in the characteristic problem from the spacetime and puts the characteristic problem on +the same footing as the standard Cauchy problem. A double null data set consists of two +null hypersurface data D and D with respective boundaries S and S, together with a non +vanishing function at the “corner”. In order for the double null data to be embedded in +some spacetime, certain compatibility conditions at S must be fulfilled. Geometrically, +they arise from the fact that in the embedded case the boundaries are identified. Con- +cretely, these conditions correspond to (i) the induced metrics, (ii) the corresponding +torsion one-forms and second fundamental forms, and (iii) the pullbacks of the ambient +Ricci tensor be the same. As we show in [13], these conditions can be written solely in +terms of the abstract data. While (i) and (iii) was already written gauge-covariantly, +the strategy we followed to obtain (ii) was to work in a very particular gauge so that +the conditions took a simplified form. Although the existence of such gauge is always +granted, defining double null data in a gauge-fixed form is not completely satisfactory +from a mathematical and geometric point of view. There may be situations where writ- +ing the data in some other gauge could simplify the problem, so giving a gauge-covariant +definition of double null data becomes necessary. +In this paper we extend the definition of double null data and make it fully gauge covari- +ant. The construction is based on the properties of (codimension one) non-degenerate +submanifolds of any given null hypersurface data. For such submanifolds we introduce +the notion of normal pair, which abstractly captures the idea of a normal vector to the +2 + +submanifold when embedded in an ambient space. It turns out that the (ambient) sec- +ond fundamental form of the submanifold along a normal vector can be written solely in +terms of the abstract data and the associated normal pair. This allows us to define, at +the abstract level, the second fundamental form of a submanifold along a normal pair. +In a similar way, one can define abstractly the notion of torsion one-forms associated to +a basis of normal pairs. It turns out that both the second fundamental form and the +torsion one-forms are gauge-covariant, so it is natural to write the compatibility condi- +tions for (ii) in terms of these objects. To achieve this one needs to “glue” abstractly +the two boundaries together in a (metric) hypersurface data sense. This is accomplished +by means of the notion of ∂-isometry, which depends on a diffeomorphism between the +boundaries as well as on a non-vanishing function that geometrically corresponds to the +scalar product of the null generators at the intersection whenever the data happens to +be embedded. Then, the gauge-covariant definition of double null data consists of two +null hypersurface data with their boundaries identified via a ∂-isometry and satisfying +the compatibility conditions. When the double null data happens to be embedded, this +notion corresponds to two transverse, null hypersurfaces, as desired. +The compatibility conditions are necessary for any double null data to be embeddable +in some ambient spacetime. The question of whether these conditions are also sufficient +arises naturally. We answer this in the affirmative by showing that given double null +data there always exists a spacetime where it can be embedded. Obviously there are +many spacetimes where a given double null data can be embedded and they are in gen- +eral not solutions of any field equations. In that sense, the compatibility conditions are +of geometric nature, and they need to be present regardless the spacetime one wants to +construct is a solution of the Einstein equations (or any other field equations) or not. +If in addition the abstract constraint equations hold, we proved in [13] that the double +null data can be embedded in a spacetime solution of the Λ-vacuum Einstein equations. +Our strategy there was to work in the so-called harmonic gauge (which is still diffeomor- +phism covariant) and solve the reduced Einstein equations from the metric data. Once +the spacetime is constructed we built new embedded data and show that (i) such embed- +ded data coincides with the original one and (ii) that the spacetime is a solution of the +Einstein equations. Thus, there is a clear hierarchy between the compatibility conditions +and the constraint equations, in the sense that the former are purely geometric and the +later depend on the field equations one wants to solve. +The other central question we want to address in this paper concerns the geometric +uniqueness of the characteristic problem at the abstract level. In the standard Cauchy +problem, it is known that two initial data sets (Σ, h, K) and (Σ′, h′, K′) such that (Σ, h) +and (Σ′, h′) are isometric and the isometry maps K′ into K, give rise to two isometric +spacetimes (M, g) and (M′, g′) [16]. To find a result of this type in the characteristic +case we need to give a notion of isometry between two abstract initial data. To do that +it is essential to take into account the gauge freedom present in the hypersurface data +formalism, since two abstract hypersurface data D and D′ related by a gauge transforma- +tion are indistinguishable from a geometric point of view. Hence, the notion of isometric +double null data involves a diffeomorphism between the two abstract hypersurfaces (and +their corresponding abstract tensors) as well as a a gauge transformation. With this +definition at hand, we prove that two double null data satisfying the constraint equa- +tions are isometric (in the abstract sense) if and only if the spacetimes they define are +isometric (in the standard sense). +3 + +This paper is structured as follows. +In Section 2 we recall the basic definitions and +results of hypersurface data from [11, 10], such as the notion of hypersurface data, +embeddedness and gauge transformation. We also summarize definitions and results from +[13]. In Section 3 we study non-degenerate codimension-one submanifolds of hypersurface +data, showing that the notion of second fundamental forms and torsion one-forms of a +submanifold can be defined abstractly from hypersurface data and a new object called +normal pair. In Section 4 we give a fully diffeomorphism and gauge covariant definition +of double null data, thus generalizing our previous definition in [13], and we show that +the compatibility conditions are necessary and sufficient for a double null data to be +embeddable in some spacetime. To do that we need the explicit expression of the pullback +of the constraint tensor into the boundary, which is obtained in Appendix A. Finally, +in Section 5, we study the necessary conditions for two different initial data to define +isometric spacetimes. This conditions lead to the definition of isometric double null data. +The paper finishes with the proof that two isometric double null data define isometric +spacetimes. This gives a geometric uniqueness notion of the characteristic initial value +problem in a fully abstract way. +Notation and conventions +Let H be a smooth manifold. We denote by F(H) the set of smooth real functions +on H, and by F ⋆(H) the subset of nowhere-vanishing functions. The tangent (resp. +cotangent) bundle of H is denoted by TH (resp. T ⋆H), and the set of vector fields (resp. +covector fields) is X(H) (resp. X⋆(H)). The interior contraction of a tensor T with a +vector X is ιXT := T(X, · · ·). Given a diffeomorphism φ : N −→ M and a vector +field X ∈ X(M), the pullback of X is given by φ⋆X := (φ−1)⋆X. We employ Greek +letters (µ, ν, ...) for spacetime abstract indices, small Latin letters from the beginning +of the alphabet (a, b, ...) for abstract indices on H and capital Latin letters from the +beginning of the alphabet (A, B, ...) for abstract indices on a section of H. Given a map +ϕ : M −→ N and a submanifold S of M, we denote by ϕ|S : S −→ ϕ(S) the restriction +of ϕ to S, both in the domain and the image. As usual, parenthesis (resp. brackets) +denote symmetrization (resp. antisymmetrization) of indices, and ⊗s is the symmetrized +tensor product, i.e., T1 ⊗s T2 := 1 +2 (T1 ⊗ T2 + T2 ⊗ T1). Our convention for the curvature +tensor of a connection ∇ is R(X, Y )Z = ∇X∇Y Z − ∇Y ∇XZ − ∇[X,Y ]Z, and its Ricci +tensor is the contraction between its contravariant index and its second covariant index. +In this paper all manifolds are connected unless otherwise indicated. +2 +Preliminaries +In this section we summarize the hypersurface data formalism. Details can be found in +[10, 11] and its precursor [12]. Our main interest is the characteristic case, so we shall +also recall from our previous paper [13] the necessary definitions and results for this case. +Definition 2.1. Let H be a smooth m-dimensional manifold, γ a symmetric two-covariant +tensor field, ℓ a one-form and ℓ(2) a scalar on H. A four-tuple Dmet = {H, γ, ℓ, ℓ(2)} +defines metric hypersurface data (of dimension m) provided that the symmetric two- +covariant tensor A|p on TpH × R defined by +A|p ((W, a), (Z, b)) := γ|p(W, Z) + aℓ|p(Z) + bℓ|p(W) + abℓ(2)|p +4 + +is non-degenerate at every p ∈ H. A five-tuple D = Dmet∪{Y}, where Y is a symmetric +two-covariant tensor field on H, is called hypersurface data. +Since the tensor A is non-degenerate one can define its contravariant “inverse” version +by A♯ (A ((V, a), ·) , ·) = (V, a) for every (V, a) ∈ X(H) × F(H). From A♯ one defines the +two-contravariant, symmetric tensor field P, the vector n and the scalar function n(2) on +H by means of +A♯ ((α, a), (β, b)) = P(α, β) + an(β) + bn(α) + abn(2), +for all α, β ∈ X⋆(H) and a, b ∈ F(H). Alternatively, P, n and n(2) can be defined by +γ(n, ·) + n(2)ℓ = 0, +(1) +ℓ(n) + n(2)ℓ(2) = 1, +(2) +P(ℓ, ·) + ℓ(2)n = 0, +(3) +P (γ(X, ·), ·) = X − ℓ(X)n +∀X ∈ X(H), +(4) +γ (P(α, ·), ·) = α − α(n)ℓ +∀α ∈ X⋆(M). +(5) +Despite its name, the abstract manifold H in Definition 2.1 is not a hypersurface of any +ambient space. The connection with the standard notion of hypersurface is given in the +following definition. +Definition 2.2. A metric hypersurface data Dmet = {H, γ, ℓ, ℓ(2)} is embedded in a +semi-Riemannian manifold (M, g) if there exists an embedding Φ : H ֒→ M and a +vector field ξ along Φ(H) everywhere transversal to Φ(H), called rigging, such that +Φ⋆(g) = γ, +Φ⋆ (g(ξ, ·)) = ℓ, +Φ⋆ (g(ξ, ξ)) = ℓ(2). +(6) +The hypersurface data D = {H, γ, ℓ, ℓ(2), Y} is embedded provided that, in addition, +1 +2Φ⋆ (Lξg) = Y. +(7) +As usual, we define the radical of γ by +Rad(γ) := {X ∈ X(H) such that γ(X, ·) = 0} . +It can be shown that the vector space Rad(γ)p at each p ∈ H is at most one-dimensional +[11]. Then, from equation (1), the condition n(2) = 0 is equivalent to Rad(γ) = span {n}. +When the data is embedded, the first relation in (6) together with Rad(γ) ̸= {0} imply +that Φ(H) is a null hypersurface. Motivated from this geometric picture, hypersurface +data satisfying n(2) = 0 everywhere on H will be called null hypersurface data (NHD). +A smooth submanifold S ⊂ H is called a section of H provided that every integral curve +of n intersects transversely S exactly once. +Let D be embedded hypersurface data. Given a rigging ξ, any other vector of the form +ξ′ = z(ξ + ζ), with z ∈ F ⋆(H) and ζ ∈ X(H), is again a rigging vector. At the abstract +level, this freedom is encoded in the following definition. +5 + +Definition 2.3. Let D = {H, γ, ℓ, ℓ(2), Y} be hypersurface data. Let z ∈ F ⋆(H) and +ζ ∈ X(H). The gauge transformed hypersurface data with gauge parameters (z, ζ) are +G(z,ζ) (γ) := γ, +(8) +G(z,ζ) (ℓ) := z (ℓ + γ(ζ, ·)) , +(9) +G(z,ζ) +� +ℓ(2)� := z2 � +ℓ(2) + 2ℓ(ζ) + γ(ζ, ζ) +� +, +(10) +G(z,ζ) (Y) := zY + ℓ ⊗s dz + 1 +2Lzζγ. +(11) +The gauge transformation laws (8)-(10) induce the following transformations on the +contravariant data [10] +G(z,ζ) (P) = P + n(2)ζ ⊗ ζ − 2ζ ⊗s n, +(12) +G(z,ζ) (n) = z−1(n − n(2)ζ), +(13) +G(z,ζ) +� +n(2)� += z−2n(2). +(14) +We will often employ a prime to denote the gauge transformed objects when the gauge +group element is clear. The set of gauge transformations defines a group with composition +law and inverse given by [10] +G(z1,ζ1) ◦ G(z2,ζ2) = G(z1z2,ζ2+z−1 +2 +ζ1) +(15) +G−1 +(z,ζ) = G(z−1,−zζ), +(16) +and neutral element G(1,0). Given hypersurface data we define the following two-covariant +tensor fields +F := 1 +2dℓ +(17) +and +K := n(2)Y + 1 +2Lnγ + ℓ ⊗s dn(2). +(18) +When the data is embedded the tensor K corresponds [11] to the second fundamental +form of Φ(H) w.r.t. the unique normal vector ν satisfying g(ν, ξ) = 1. As expected from +this geometric interpretation, the transformation law of K is +G(z,ζ) (K) = z−1K. +(19) +Given hypersurface data, a unique torsion-free connection ∇ exists with the following +defining properties +� +∇Xγ +� +(Z, W) = −ℓ(Z)K(X, W) − ℓ(W)K(X, Z), +(20) +� +∇Xℓ +� +(Z) = Y(X, Z) + F(X, Z) − ℓ(2)K(X, Z). +(21) +Under a gauge transformation with gauge parameters (z, ζ), the connection ∇ transforms +as [13] +G(z,ζ) +� +∇ +� += ∇ + ζ ⊗ K. +(22) +The combination Y + F will appear frequently below, so we give it a name, +Π := Y + F. +(23) +6 + +The tensor Π has the following interesting property [13, Eq. (36)] +Π(n, X) − Π(X, n) = (Lnℓ) (X). +(24) +The transformation law of Π(·, n) is [13, Eq. (38)] +Π′(·, n′) = Π(·, n) − K(·, ζ) + d log |z|. +(25) +A consequence of (20)-(21) together with (1)-(4) is [10] +∇Xn = P +� +K(X, ·) − n(2)Π(X, ·), · +� +− +� +Π(X, n) + n(2)X +� +ℓ(2)�� +n. +(26) +For null hypersurface data (n(2) = 0) the previous equation takes the simpler form +∇Xn = P (K(X, ·), ·) − Π(X, n)n. +(27) +The connection ∇ has the following geometric interpretation in the embedded case. +Given Dmet with embedding Φ and rigging ξ in an ambient manifold (M, g) with Levi– +Civita connection ∇ and X, Z ∈ X(H), the relation between ∇ and ∇ is [10, 12] +∇XZ = ∇XZ − K(X, Z)ξ, +(28) +where we have identified vector fields in X(H) and their image under Φ⋆. This slight +abuse of notation will be used repeatedly from now on. +The rest of this section is devoted to summarizing the results of characteristic hypersur- +face data developed in [13] needed in this paper. +Definition 2.4. Let D = {H, γ, ℓ, ℓ(2), Y} be hypersurface data of dimension m. We +say that the set D is “characteristic hypersurface data” (CHD) provided that +1. Rad(γ) ̸= {0} and γ is semi-positive definite. +2. There exists u ∈ F(H) satisfying λ := n(u) ̸= 0. Such functions are called “folia- +tion functions” (FF). +3. The leaves Su := {p ∈ H : u(p) = u} are all diffeomorphic. +Henceforth, we will assume H to have boundary of the form ∂H = {u = u0}. Given CHD, +the existence of such foliation function allows us to define a tangent space decomposition +at every p ∈ H of the form +TpH = TpSu(p) ⊕ span {n|p} , +(29) +where TpSu(p) = {X ∈ TpH such that X(u) = 0}. This decomposition induces an- +other on the cotangent space given by T ⋆ +p H = T ⋆ +p Su(p) ⊕ span {du|p}. Then the tangent +(resp. cotangent) bundle decomposes as the direct sum TH = TS ⊕ span {n} (resp. +T ⋆H = T ⋆S ⊕ span {du}), where TS = � TqSu, and therefore every X ∈ X(H) can be +written uniquely as X = X∥ + σXn with X∥ ∈ X(S) and σX ∈ F(H). A vector field X +is said to be tangent to the foliation provided σX = 0. +Let D be CHD, u a foliation function and i : Su ֒→ H the inclusion of each leaf onto H. +For each Su we define h := i⋆γ, the one-form +ℓ∥ := i⋆ℓ ∈ X⋆(Su), +(30) +7 + +and the vector +ℓ♯ := h♯(ℓ∥, ·). +(31) +It is immediate to show that h is a positive definite metric. The transformation laws of +ℓ∥ and ℓ♯ follow directly from that of ℓ (see (9)) +ℓ′ +∥ = z +� +ℓ∥ + h(ζ∥, ·) +� +, +(32) +ℓ′♯ = z(ℓ♯ + ζ∥). +(33) +In [13] the following set of so-called “foliation tensors” were introduced. By construction +they are intrinsic to the leaves of the foliation. +Definition 2.5. Let D be CHD and u a foliation function. We define the “foliation +tensors” χ, Υ, τ and η on each leaf Su by +χ := i⋆K, +Υ := i⋆Y, +τ := i⋆ (Π(·, n) + ιℓ♯K) +η := −τ − d(log |λ|). +In the embedded case, χ is the second fundamental form of Su along the normal vector +ν. The tensors Υ and η do not admit such a neat geometric interpretation. However, +in an appropriate gauge (namely such that the rigging is null and normal to Su), Υ +coincides with the null second fundamental form along ξ and η is the torsion one-form +of Su w.r.t the null basis {ν, ξ}. Finally we recall that Y(n, n) measures the deviation +of ν from being geodesic (this follows exclusively from equations (21), (28) and therefore +it does not depend on the gauge or on the existence of a foliation on H). We refer the +reader to [13, Sect. 5] for the details. +As shown in [13], given embedded null hypersurface data in a spacetime (M, g), all +tangential components of the ambient Ricci tensor on the hypersurface can be written +in terms of the abstract data. This allowed us to define the abstract constraint tensor +Rab by +Rab := +� +γafR +f +cbd + 2∇[d +� +Kb]cℓa +� ++ 2ℓ(2)Kc[bKd]a +� +P cd +(34) +− +� +ℓdR +d +bac + 2ℓ(2)∇[cKa]b + Kb[a∇c]ℓ(2) + ℓdR +d +abc + 2ℓ(2)∇[cKb]a + Ka[b∇c]ℓ(2)� +nc, +where R +a +bcd is the curvature tensor of ∇. As shown in [13], this tensor satisfies +R = Φ⋆ Ric, +(35) +where Ric is the Ricci tensor of (M, g). This property suggests that R is gauge invariant +and, indeed, in Theorem 4.6 in [13] we have established +G(z,ζ) (R) = R +∀(z, ζ) ∈ F ⋆(H) × X(H). +(36) +By the geometric interpretation of this tensor, the condition R = 0 can be thought as +the vacuum constraint equations on a null hypersurface H. These equations are fully +diffeomorphism and gauge covariant and do not assume the presence of any foliation, +thus generalizing the so-called null structure equations. Recall that the hypersurface +data formalism allows us to write these constraint equations even though there is no +metric on H, thanks to the existence of the abstract connection ∇. +8 + +In Sect. 6 of [13] we introduced a fully covariant gauge which translates the harmonic +condition on the ambient coordinates into an abstract condition on the data. Given +a set of independent functions {xa} on H and a (local) basis {ec} of X(H), we define +Bca := ec(xa) and Bca as its inverse. Since, for each value of a, Bca is a vector so it is +V c := Bca□Pxa, where □P f := P ab∇a∇bf for every scalar f. This vector allows us to +define the so-called harmonic gauge, which one can show it exists and is unique up to +the choice of the gauge parameters at a given “initial” section. We refer the reader to +Theorem 6.2 of [13] for the proof. +Theorem 2.6. Let D be null hypersurface data admitting a section S ⊂ H, {xa} a set +of m functionally independent functions, V c := Bca□P xa and trP K := P abKab. Then +there exists a class of gauges satisfying the following conditions on H, +trP K − 2Y(n, n) = 0, +(37) +2P +� +∇nℓ, · +� ++ n +� +ℓ(2)� +n = V. +(38) +The set of transformations keeping the previous conditions invariant is parametrized by +(z0, ζ0) ∈ F ⋆(S) × X(H). Any gauge satisfying (37) and (38) will be called “harmonic +gauge” (HG). +The name “harmonic” is motivated by the following property. Given embedded CHD +in a spacetime (M, g) with rigging ξ and embedding Φ, we can extend the functions xa +off Φ(H) by ξ(xa) = 0. Moreover, we can introduce another function u on M such that +u|H = 0 and ξ(u) = 1 on Φ(H). As shown in [13], the data is written in the harmonic +gauge if and only if the functions {u, xa} satisfy □gxa = □gu = 0 on Φ(H). +One can exploit the remaining gauge freedom in the class of gauges of Theorem 2.6 to +fix the values of ℓ(2) and ℓ∥ to zero in any section S (see [13, Cor. 4.4]). +Lemma 2.7. Let D be null hypersurface data admitting a section S ⊂ H, {xa} a set +of m functionally independent functions and V c := Bca□Pxa. Then there exists a class +of gauges satisfying (37)-(38) on H as well as ℓ(2) = 0, ℓ∥ = 0 on S. +The set of +transformations keeping the previous conditions invariant is parametrized by z0 ∈ F ⋆(S). +Let {xa} = {u, xA} be functions on H satisfying (i) n(u) ̸= 0, (ii) n(xA) = 0 and (iii) +{xA} is a coordinate system on S. When the data is written in the gauge of Lemma 2.7 +w.r.t {xa}, conditions (38) can be rewritten as equations involving the Y tensor (see [13, +Prop. 7.13]). +Proposition 2.8. Let D be null hypersurface data admitting a section S written in the +gauge of Lemma 2.7 w.r.t some functions xa = {u, xA} satisfying conditions (i), (ii) and +(iii) above. Then the following equations hold at S +trh Υ := hABΥAB = n +� +ℓ(2)� +, +(39) +2 (Lnℓ + Π(·, n)) (gradhxA) = □hxA. +(40) +3 +Non-degenerate submanifolds +In this section we study embedded non-degenerate submanifolds in the context of hy- +persurface data. For the purposes of this paper we restrict ourselves to the null case but +9 + +without fixing a priori the topology of H. Let D = {H, γ, ℓ, ℓ(2), Y} be null hypersurface +data of dimension m and Σ an (m − 1)-dimensional manifold. We say that i : Σ ֒→ H +is a non-degenerate submanifold (of codimension one) of D provided that h := i⋆γ is +non-degenerate. The idea is to identify, in terms of hypersurface data, a suitable notion +of the torsion and second fundamental form of Σ. We start by assuming that D is em- +bedded null hypersurface data with rigging ξ and embedding Φ in a spacetime (M, g) +with Levi-Civita connection ∇. Given a vector field t along Φ (i (Σ)) everywhere normal +to Σ, there exist a function C ∈ F(Σ) and a vector field T ∈ X(H) such that +t = Cξ + Φ⋆T. +(41) +By (6), the condition that t is normal to Σ is equivalent to +Cℓ(X) + γ(T, X) = 0 +∀X ∈ X(Σ). +Given X, Y ∈ X(Σ), the (ambient) second fundamental form IIt of Σ along the normal +vector t can be defined as +IIt(X, Y ) = 1 +2 (Ltg) (X, Y ). +Substituting the expression (41) and employing the simple formula +LfV g = fLV g + 2df ⊗s g(V, ·) +(42) +valid for any function f and vector field V , +IIt(X, Y ) = 1 +2 (LΦ⋆Tg) (X, Y ) + 1 +2C (Lξg) (X, Y ) + (dC ⊗s g(ξ, ·))(X, Y ). +Inserting the expressions of Y and ℓ in the context of embedded hypersurface data (see +Def. 2.2) and using Φ⋆ (LΦ⋆Tg) = LT (Φ⋆g) = LTγ, one finally gets +IIt(X, Y ) = 1 +2 (LTγ) (X, Y ) + CY(X, Y ) + 1 +2 (X(C)ℓ(Y ) + Y (C)ℓ(X)) , +(43) +which is an expression only depending on (abstract) hypersurface data. In addition to the +second fundamental form, the geometry of submanifolds of (ambient) codimension two +also consist of the torsion one-form. Given a basis of normal vectors {ti = Ciξ + Φ⋆Ti}, +i = 1, 2, the (ambient) torsion of Σ w.r.t. this basis is the set of one-forms Tij given by +Tij(X) = g(ti, ∇Xtj) +∀X ∈ X(Σ). +(44) +As in the case of the second fundamental form, this object can be also rewritten in terms +of hypersurface data. Indeed, from (41) and (28) +∇Xtj = ∇X (Cjξ + Φ⋆Tj) += (X(Cj) − K(X, Tj)) ξ + Cj∇Xξ + ∇XTj. +(45) +Then, using (6) +g(ξ, ∇Xtj) = (X(Cj) − K(X, Tj)) ℓ(2) + 1 +2CjX(ℓ(2)) + ℓ(∇XTj), +(46) +and +g(Φ⋆Ti, ∇Xtj) = (X(Cj) − K(X, Tj)) ℓ(Ti) + Cjg(Φ⋆Ti, ∇Xξ) + γ(Ti, ∇XTj). +10 + +From equations (28), (21) and the definition of ℓ in the context of embedded data (Def. +2.2), +g(Φ⋆Ti, ∇Xξ) = X (g(Φ⋆Ti, ξ)) − g (∇X(Φ⋆Ti), ξ) += X (g(Φ⋆Ti, ξ)) − g +� +∇XTi, ξ +� ++ K(X, Ti)g(ξ, ξ) += X (ℓ(Ti)) − ℓ +� +∇XTi +� ++ ℓ(2)K(X, Ti) += +� +∇Xℓ +� +(Ti) + ℓ(2)K(X, Ti) += Π(X, Ti). +Then, +g(Φ⋆Ti, ∇Xtj) = (X(Cj) − K(X, Tj)) ℓ(Ti) + CjΠ(X, Ti) + γ(Ti, ∇XTj). +(47) +Introducing (45) into (44) and employing (46) and (47), +Tij(X) = g(Ciξ + Φ⋆Ti, ∇Xtj) += +� +ℓ(Ti) + Ciℓ(2)� +(X(Cj) − K(X, Tj)) + Ciℓ +� +∇XTj +� ++ γ(Ti, ∇XTj) + 1 +2CiCjX +� +ℓ(2)� ++ CjΠ(X, Ti). +(48) +The computation above suggests introducing a fully abstract notion of normal vector to +the submanifold. This notion is called “normal pair” (NP) and it is defined as follows. +Definition 3.1. Let D = {H, γ, ℓ, ℓ(2), Y} be hypersurface data and i : Σ ֒→ H a non- +degenerate submanifold. A normal pair t := {T, C} consists of a vector field T ∈ X(H) +along i(Σ) and a function C ∈ F(Σ) satisfying +Cℓ(X) + γ(T, X) = 0 +∀X ∈ X(Σ). +(49) +Since Σ is a codimension one submanifold of H, Σ admits exactly two linearly indepen- +dent NPs. Indeed, equation (49) can be written in terms of the tensor A (see Def. 2.1) +as +A ((T, C), (X, 0)) = 0 +for every X ∈ X(Σ) along i(Σ). Since A|p is non-degenerate at every p ∈ Σ, the (m+1)- +dimensional vector space TpH × R can be decomposed as +TpH × R = TpΣ ⊕ (TpΣ)⊥ . +Since TpΣ has dimension (m − 1) it follows that (TpΣ)⊥ is two-dimensional and hence Σ +admits exactly two linearly independent NPs. Motivated by (43), given a normal pair +t = {T, C}, the second fundamental form of Σ along t is the tensor field on Σ defined by +Kt(X, Y ) := 1 +2 (LTγ) (X, Y ) + CY(X, Y ) + 1 +2 (X(C)ℓ(Y ) + Y (C)ℓ(X)) , +(50) +for X, Y ∈ X(Σ). In the embedded case, given a normal pair t = {C, T} we can associate +to it the (ambient) vector field t[t] defined by +t[t] := Cξ + Φ⋆T. +By construction, t[t] is normal to Φ(i(Σ)), since g(t[t], X) = Cℓ(X) + γ(T, X) = 0 for +every X ∈ X(Σ). Comparing the expression of the (ambient) second fundamental form +IIt[t] of Φ (i(Σ)) in (43) with the abstract Kt in definition (50), the following result follows. +11 + +Proposition 3.2. Let D = {H, γ, ℓ, ℓ(2), Y} be embedded hypersurface data on a space- +time (M, g) with embedding Φ and rigging ξ, and let Σ be a non-degenerate submanifold +of H. Let t = {T, C} be a NP of Σ and let t = Cξ + Φ⋆T be its associated (ambient) +vector field. Then, +Kt(X, Y ) = IIt[t](X, Y ) +for every X, Y ∈ X(Σ). +As expected from the spacetime interpretation, Kt has the following properties. +Lemma 3.3. Let t = {T, C} be a normal pair, Kt the second fundamental form as in +(50) and f a scalar function. Then, +Kft = fKt. +Proof. Directly from the definition (50) and the formula (42), +Kft(X, Y ) = 1 +2 (LfTγ) (X, Y ) + fCY(X, Y ) + (d (fC) ⊗s ℓ) (X, Y ) += fKt(X, Y ) + (df ⊗s γ(T, ·)) (X, Y ) + (Cdf ⊗s ℓ) (X, Y ) += fKt(X, Y ) + (df ⊗s (γ(T, ·) + Cℓ)) (X, Y ). +Since t is a normal pair, the term in parenthesis in the last line vanishes when contracted +with tangent vectors, and thus result follows. +Lemma 3.4. Let {ti} be a basis of normal pairs and ˆti = Ωj +itj a change of basis, with +Ωj +i ∈ F(Σ). Then, +K +ˆti = Ωj +iKti. +Proof. By the previous Lemma it suffices to show that Kt1+t2 = Kt1 + Kt2 for every pair +of NPs t1 = (T1, C1) and t2 = (T2, C2). Directly from the definition of K in (50), +Kt1+t2(X, Y ) = 1 +2 (LT1+T2γ) (X, Y ) + (C1 + C2)Y(X, Y ) ++ 1 +2 (X(C1 + C2)ℓ(Y ) + Y (C1 + C2)ℓ(X)) += Kt1(X, Y ) + Kt2(X, Y ). +Finally, Kt1+t2 = Kt1 + Kt2 together with Kft = fKt proves K +ˆti = Ωj +iKti. +The spacetime picture also motivates the following definition. +Definition 3.5. Let D = {H, γ, ℓ, ℓ(2), Y} be hypersurface data and i : Σ ֒→ H a non- +degenerate submanifold. Let (z, ζ) be a gauge group element. The gauge transformation +of a normal pair t = {T, C} of Σ is defined as +G(z,ζ)(t) := {T ′ = T − Cζ, C′ = z−1C}. +Note that t′ = G(z,ζ)(t) is still a normal pair, since +C′ℓ′(X) + γ(T ′, X) = Cℓ(X) + Cγ(ζ, X) + γ(T, X) − Cγ(ζ, X) = 0. +12 + +Moreover, Def. 3.5 is a realization of the gauge group. Indeed, let (z1, ζ1), (z2, ζ2) be +gauge parameters. On the one hand, +G(z1,ζ1)G(z2,ζ2){T, C} = G(z1,ζ1) +� +T − Cζ2, z−1 +2 C +� += +� +T − (z−1 +2 ζ1 + ζ2)C, z−1 +1 z−1 +2 C +� +, +and on the other, using (15), +G(z1,ζ1)G(z2,ζ2){T, C} = G(z1z2,ζ2+z−1 +2 +ζ1) {T, C} = +� +T − (z−1 +2 ζ1 + ζ2)C, z−1 +1 z−1 +2 C +� +. +As expected from the spacetime picture, the second fundamental form along a normal +pair is gauge invariant. +Lemma 3.6. Let D be hypersurface data, Σ a non-degenerate submanifold and t a normal +pair of Σ. Then Kt is gauge invariant. +Proof. Let t = {T, C} and t′ = {T ′ = T − Cζ, C′ = z−1C}. Using the transformation +laws (9) and (11) (we drop the argument (X, Y ) in order not to overload the notation) +Kt′ = 1 +2LT ′γ + C′Y′ + dC′ ⊗s ℓ′ += 1 +2LTγ − 1 +2CLζγ − dC ⊗s γ(ζ, ·) + CY + z−1Cℓ ⊗s dz + 1 +2CLζγ ++ z−1Cdz ⊗s γ(ζ, ·) + dC ⊗s (ℓ + γ(ζ, ·)) + zCdz−1 ⊗s (ℓ + γ(ζ, ·)) += 1 +2LTγ + CY + dC ⊗s ℓ += Kt, +where in the second line we used the formula (42). +Let i : Σ ֒→ H be a non-degenerate submanifold. Following the notation in (30)-(31) in +the context of CHD, we define the one-form ℓ∥ ∈ X⋆(Σ) and the vector ℓ♯ ∈ X(Σ) by +means of +ℓ∥ := i⋆ℓ, +ℓ♯ := h♯(ℓ∥, ·). +(51) +In the next proposition we identify a particular relevant normal pair. +Proposition 3.7. Let D = {H, γ, ℓ, ℓ(2), Y } be null hypersurface data and i : Σ ֒→ H a +non-degenerate submanifold. Define the vector θ along i(Σ) by +θ := −1 +2 +� +ℓ(2) − h +� +ℓ♯, ℓ♯�� +n − i⋆ℓ♯. +(52) +Then, tθ := (θ, 1) is a normal pair of Σ. Furthermore, tθ is A-null, i.e., A(tθ, tθ) = 0. +Proof. For any X ∈ X(Σ), +A ((i⋆X, 0), (θ, 1)) = ℓ(i⋆X) + γ(i⋆X, θ) = ℓ∥(X) − h(X, ℓ♯) = 0, +where we used γ(n, ·) = 0. Hence, tθ is a NP. Moreover, +A ((θ, 1), (θ, 1)) = γ(θ, θ) + 2ℓ(θ) + ℓ(2) = h(ℓ♯, ℓ♯) − ℓ(2) + h +� +ℓ♯, ℓ♯� +− 2ℓ∥(ℓ♯) + ℓ(2) = 0, +where now we used γ(n, ·) = 0 and ℓ(n) = 1. +13 + +Remark 3.8. Observe that the normal pair tn := (n, 0) is linearly independent of tθ and +also satisfies A(tn, tn) = 0. In fact, since the space of normal pairs is two-dimensional +and +A(tn, tθ) = γ(n, θ) + ℓ(n) = 1, +it follows that any normal pair which is also A-null lies either in span{tn} or span{tθ}. +From equation (48) the following abstract definition of the torsion one-forms is motivated. +Definition 3.9. Let D = +� +H, γ, ℓ, ℓ(2), Y +� +be null hypersurface data and {ti = (Ti, Ci)} +with i = 1, 2, a basis of normal pairs. The torsion one-forms of Σ w.r.t this basis are +the tensors on Σ defined by +ℸij(X) := +� +ℓ(Ti) + Ciℓ(2)� +(X(Cj) − K(X, Tj)) + Ciℓ +� +∇XTj +� ++ γ(Ti, ∇XTj) + 1 +2CiCjX +� +ℓ(2)� ++ CjΠ(X, Ti) +for every X ∈ X(Σ). We will also employ the notation ℸ(ti, tj) when ℸij causes confusion. +Let {ti, tj} be a basis of normal pairs and let ti[ti], tj[tj] be their associated (ambient) +vector fields. Comparing the expression of the (ambient) torsion one-forms Tij of Φ (i(Σ)) +in (44) with the abstract ℸij in Definition 3.9, the following result follows. +Proposition 3.10. Let D = {H, γ, ℓ, ℓ(2), Y} be embedded hypersurface data on a space- +time (M, g) with embedding Φ and rigging ξ, and let Σ be a non-degenerate submanifold +of H with a basis of normal pairs {ti}. Then, +ℸij(X) = Tij(X) +for every X ∈ X(Σ), where Tij are the ambient torsion one-forms w.r.t the basis {ti[ti]}. +Since Σ is a codimension-one submanifold of H one has in principle four different torsion +one-forms ℸij. However, not all them are independent. +Proposition 3.11. Let D = +� +H, γ, ℓ, ℓ(2), Y +� +be null hypersurface data, {(Ti, Ci)} a +basis of normal pairs and +Mij := A ((Ti, Ci), (Tj, Cj)) = γ(Ti, Tj) + Ciℓ(Tj) + Cjℓ(Ti) + CiCjℓ(2). +(53) +Then, +ℸij + ℸji = dMij. +Proof. Directly from the definition of ℸ, +ℸij(X) + ℸji(X) = X +� +CiCjℓ(2)� ++ 2 +� +X(C(i) − K(X, T(i) +� +ℓ(Tj)) − 2ℓ(2)C(iK(X, Tj)) ++ 2C(iℓ +� +∇XTj) +� ++ 2γ +� +T(i, ∇XTj) +� ++ 2C(iΠ(X, Tj)). +(54) +Using equation (20), +2γ +� +T(i, ∇XTj) +� += X (γ(Ti, Tj)) − +� +∇Xγ +� +(Ti, Tj) = X (γ(Ti, Tj)) + 2K(X, T(i)ℓ(Tj)). +(55) +From (21), +2C(iℓ +� +∇XTj) +� += 2X +� +C(iℓ(Tj)) +� +− 2X(C(i)ℓ(Tj)) − 2C(i +� +∇Xℓ +� +(Tj)) += 2X +� +C(iℓ(Tj)) +� +− 2X(C(i)ℓ(Tj)) − 2C(iΠ(X, Tj)) + 2ℓ(2)C(iK(X, Tj)). +(56) +Inserting (55) and (56) into (54) yields the result. +14 + +In fact, since Σ is a codimension one submanifold of H, there is only one independent +torsion one-form on Σ, that can be taken to be ℸ12. In the embedded case, given two +normal pairs ti and tj, the function Mij = A (ti, tj) as defined in (53) corresponds to the +scalar product g(ti, tj), where ti and tj are the ambient vector fields associated to ti and +tj, respectively. As expected from this geometric interpretation, the functions Mij are +gauge invariant, as we show next at the abstract level. +Lemma 3.12. Let {ti} be a basis of normal pairs of a non-degenerate submanifold Σ +and Mij := A (ti, tj). Then, the functions Mij are gauge invariant. +Proof. Let (z, ζ) be gauge parameters. Directly from (53) and Def. 3.5, +M′ +ij = A′ � +(Ti − Ciζ, z−1Ci), (Tj − Cjζ, z−1Cj) +� += γ (Ti − Ciζ, Tj − Cjζ) + z−1Ciℓ′ (Tj − Cjζ) + z−1Cjℓ′ (Ti − Ciζ) + z−2CiCjℓ(2)′ += γ (Ti, Tj) + Ci +� +z−1ℓ′(Tj) − γ(ζ, Tj) +� ++ Cj +� +z−1ℓ′(Ti) − γ(ζ, Ti) +� ++ CiCj +� +z−2ℓ(2)′ + γ(ζ, ζ) − 2z−1ℓ′(ζ) +� += γ(Ti, Tj) + Ciℓ(Tj) + Cjℓ(Ti) + CiCjℓ(2) += Mij, +where in the fourth line we used (8)-(10). +As expected from the geometric interpretation of the ambient torsion one-forms as the +connection coefficients of the normal bundle connection, the transformation of the ab- +stract torsion one-forms under a change of basis of normal pairs is as follows. +Proposition 3.13. Let {ti} be a basis of normal pairs of Σ and ˆtj = Ωi +jti with Ωi +j ∈ F(Σ) +a change of basis. Then, +�ℸij = Ωk +i Ωl +jℸkl + Ωk +i Mkl dΩl +j. +(57) +Proof. Writing ti = (Ti, Ci) and ˆti = ( �Ti, �Ci), the change of basis gives +�Ci = Ωk +i Ck, +�Ti = Ωk +i Tk. +Inserting them in the expression of ℸij and noting that all the terms are multilinear +except for X( �Cj) and ∇X �Tj, which become +X( �Cj) = Ωl +jX(Cl) + ClX(Ωl +j), +∇X �Tj = Ωl +j∇XTl + X(Ωl +j)Tl, +one immediately gets +�ℸij(X) = Ωk +i Ωl +jℸkl + Ωk +i +� +CkClℓ(2) + Ckℓ(Tl) + Clℓ(Tk) + γ(Tk, Tl) +� +X +� +Ωl +j +� +, +which is (57) after recalling the definition of Mkl in (53). +Corollary 3.14. Let t1 and t2 two linearly independent normal pairs and f a scalar +function. Under a change of basis {t1, t2} �−→ {t′ +1 = f −1t1, t′ +2 = ft2}, the torsion one- +form ℸ(t1, t2) transforms as +�ℸ12 = ℸ12 + M12d log |f|. +In the next proposition we show that the abstract torsion one-forms are gauge invariant. +15 + +Proposition 3.15. Let D be hypersurface data, Σ a non-degenerate submanifold and +{ti = (Ti, Ci)} a basis of normal pairs. +Then, the torsion one-forms ℸij are gauge +invariant. +Proof. First we show that if ℸij are gauge invariant in a particular basis of normal pairs, +they are also invariant in any basis. Let {ti} (with i = 1, 2) be a basis of normal pairs +and let ˆtj = Ωi +jti with Ωi +j ∈ F(Σ) be a change of basis. From the transformation law in +Def. 3.5, +�Cj = Ωi +jCi, +�Tj = Ωi +jTi +=⇒ +z−1 �Cj = Ωi +jz−1Ci, +�Tj − �Cjζ = Ωi +j (Ti − Ciζ) , +which means that the gauge transformed basis {t′ +i} and {ˆt′ +i} are related by ˆt′ +j = Ωi +jt′ +i. +Thus, the functions Ωi +j are gauge invariant. Moreover, from Lemma 3.12 the functions +Mij are gauge invariant too. Then, from Proposition 3.13, if ℸij is gauge invariant, so +it is �ℸij. Hence, it suffices to show the statement in the basis {tn, tθ}, where the only +non-zero torsion one-forms are given by +ℸnθ(X) = −ℸθn(X) = −K(X, θ) + Π(X, n). +From Def. 3.5, given gauge parameters (z, ζ) the transformed basis of normal pairs is +G (tn) = (n, 0) and G (tθ) = (θ − ζ, z−1). Hence, applying Def. 3.9 to the new gauge, +G (ℸnθ) (X) = z +� +X(z−1) − K′(X, θ − ζ) +� ++ z−1Π′(X, n) += −X (log |z|) − K(X, θ) + K(X, ζ) + Π′(X, n′) += −X (log |z|) − K(X, θ) + K(X, ζ) + Π(X, n) + X (log |z|) − K(X, ζ) += ℸnθ(X), +where in the third line we used the transformation law of Π(·, n) in (25). +Remark 3.16. In the context of CHD the leaves {iu : Su ֒→ H} are non-degenerate +submanifolds of H. Then, it is natural to connect the geometry of the foliation in a CHD +with the tools developed in this Section. Let D be CHD, u a foliation function and Su +any leaf. By Remark 3.8, tn = (n, 0) and tθ = (θ, 1) constitute a basis of A-null, normal +pairs of Su. Using (50) and Definition 2.5, we have Ktn = χ and Ktθ = Υ + Θ, where +Θ := 1 +2i⋆ +u (Lθγ) . +(58) +From the definition of the torsion one-form (Def. 3.9) one also has +ℸnθ(X) = −K(X, θ) + Π(X, n) = K(X, ℓ♯) + Π(X, n) = τ(X). +Thus, the tensors χ and Υ+Θ are the second fundamental forms w.r.t the normal pairs +tn = (n, 0) and tθ = (θ, 1), respectively, whereas the tensor τ is the torsion one-form +w.r.t the basis {tn, tθ}. +4 +Gauge-covariant compatibility conditions +In Section 7 of [13] we studied the necessary conditions that two CHD must fulfil in +order to be simultaneously embedded in the same spacetime with common spacelike +boundary. However, these compatibility conditions were obtained in a particular gauge +16 + +where strong restricting conditions were imposed at the respective boundaries. In this +Section we generalize the compatibility conditions and write them employing the tools +developed in the previous section, which allows us to define the notion of double null +data in a fully gauge-covariant way. We start introducing the key notion that will allow +us to do that. +Definition 4.1. Let D and D be two null hypersurface data with boundaries S := ∂H +and S := ∂H and φ : S −→ S a diffeomorphism between them. An invertible linear map +Ψp : TpH ⊕ R −→ Tφ(p)H ⊕ R +is called a ∂-isometry between CHD at p ∈ S provided that1 +1. Ψp|TpS ((X, 0)) = φ⋆|p(X) for all X ∈ X(S), +2. Ψ⋆ +pAφ(p) = Ap. +If conditions (1) and (2) hold at every p ∈ S we say that D and D are ∂-isometric. +Since S and S are non-degenerate codimension one submanifolds of H and H, we can +talk about normal pairs on S and S. Given a NP t = {T, C} of S, the image of t under +Ψ is still a normal pair on S, since φ is a diffeomorphism and +0 = A (t, (X, 0)) = (Ψ⋆A) (t, (X, 0)) = A (Ψ(t), (φ⋆X, 0)) +∀X ∈ X(H). +Moreover, if a normal pair t of S is A-null, the normal pair Ψ(t) of S is A-null, since +A (Ψ(t), Ψ(t)) = A (t, t) = 0. +In the following proposition we show that given a diffeomorphism φ : S −→ S and a +non-vanishing function µ ∈ F ⋆(S), a natural ∂-isometry can be constructed. +Proposition 4.2. Let D and D be two NHD and φ : S −→ S a diffeomorphism. For +each µ ∈ F ⋆(S), there exists a unique ∂-isometry Ψ : TH ⊕ F(S) −→ TH ⊕ F(S) +determined by the condition A ((n, 0), Ψ(n, 0)) = µ. +Proof. The existence and uniqueness proof will be constructive, i.e., we will impose +conditions (1) and (2) in Def. +4.1 and show that there is a unique candidate. +We +will then check that this candidate is indeed a ∂-isometry. Let X ∈ X(S). We define +Ψ ((X, 0)) := (φ⋆X, 0). In order to define the image of (n, 0) and (θ, 1) under Ψ first +observe that since (n, 0) and (θ, 1) are A-null, Ψ ((n, 0)) and Ψ ((θ, 1)) are A-null too, and +since A ((n, 0), (θ, 1)) = 1 (see Remark 3.8) we necessarily have Ψ ((n, 0)) ∈ span {(θ, 1)} +and Ψ ((θ, 1)) ∈ span {(n, 0)}. Imposing the condition A ((n, 0), Ψ(n, 0)) = µ the only +option is +Ψ ((n, 0)) = µ(θ, 1). +(59) +To complete the determination of Ψ we only need to find Ψ ((θ, 1)), which we already +know is of the form Ψ ((θ, 1)) = α(n, 0). The proportionality function α is obtained from +A ((n, 0), (θ, 1)) = 1 and its underlined version after imposing item (2) of Def. 4.1 as +follows +1 = A ((θ, 1), (n, 0)) = A (Ψ ((θ, 1)) , Ψ((n, 0))) = αµA ((n, 0), (θ, 1)) = αµ. +1Given V, W ∈ TpH ⊕ R, we define +� +Ψ⋆Aφ(p) +� +(V, W) := Aφ(p) (Ψ(V ), Ψ(W)). +17 + +Since µ ̸= 0 by hypothesis we conclude +Ψ ((θ, 1)) = µ−1(n, 0). +(60) +In matrix notation, the expression of Ψ in the decomposition TS ⊕ span {(n, 0), (θ, 1)} +(and the corresponding one in the image) is2 +Ψ = + + +(φ⋆) +0 +0 +0 +0 +µ−1 +0 +µ +0 + + . +(61) +It is now straightforward to check that this candidate to be a ∂-isometry is indeed +a ∂-isometry. +Indeed, conditions (1) and (2) of Def. +4.1 hold by construction and +by expression (61) and the fact that φ is a diffeomorphism, we conclude that Ψ is +invertible. +In the proof of Prop. 4.2 we found the expression of Ψ w.r.t decompositions TpS ⊕ +span {(n, 0), (θ, 1)} and Tφ(p)S ⊕ span {(n, 0), (θ, 1)}, namely +Ψ = + + +(φ⋆) +0 +0 +0 +0 +µ−1 +0 +µ +0 + + . +(62) +In some situations it may be interesting to have an expression for Ψ in a more general +basis. +Proposition 4.3. Let D and D be NHD, φ : S −→ S a diffeomorphism and Ψ the unique +∂-isometry satisfying A ((n, 0), Ψ(n, 0)) = µ ̸= 0. Let v, v be transverse vectors to S and +S, respectively. Then the linear map Ψ w.r.t decompositions TS ⊕ span {(v, 0), (0, 1)} +and TS ⊕ span {(v, 0), (0, 1)} is given by +Ψ = + + +(φ⋆) +φ⋆v∥ − σµ +� +ασ−1v∥ + ℓ♯� +φ⋆ℓ♯ + µαℓ♯ + σ−1(µαα − µ−1)v∥ +0 +µσασ−1 +σ−1(µ−1 − µαα) +0 +µσ +−µα + + , +(63) +where σ ∈ F ⋆(S), σ ∈ F ⋆(S), v∥ ∈ X(S) and v∥ ∈ X(S) are univocally defined by the +decompositions v = σn + v∥ and v = σn + v∥, and where we introduce the functions +α = −1 +2 +� +ℓ(2) − h(ℓ♯, ℓ♯) +� +and α = −1 +2 +� +ℓ(2) − h(ℓ♯, ℓ♯) +� +to simplify the notation. +Proof. We only need to compute the second and third columns. +Since Ψ ((v, 0)) = +σΨ ((n, 0)) + Ψ +� +(v∥, 0) +� +, employing (59) together with (52), namely θ = αn − ℓ♯ (we +omit the i⋆ in order not to overload the notation), +Ψ ((v, 0)) = σµ(θ, 1) + (φ⋆v∥, 0) = +� +σµασ−1(v − v∥) − σµℓ♯ + φ⋆v∥, σµ +� +, +and hence the second column of (63) follows. The third column requires computing +Ψ ((0, 1)). Decomposing (0, 1) = (θ, 1) − (αn, 0) + (ℓ♯, 0) and using (62) yields +Ψ(0, 1) = +� +µ−1n − µαθ + φ⋆ℓ♯, −µα +� += +� +µ−1σ−1(v − v∥) − µαασ−1(v − v∥) + µαℓ♯ + φ⋆ℓ♯, −µα +� +, +so the third column of (63) is established. +2When a map is written in matrix notation we follow the convention that the entries of the i-th +column are the coefficients of the image of the i-th vector in the basis. +18 + +Next we study how the map Ψ changes under gauge transformations on D and D. +Proposition 4.4. Let D and D be null hypersurface data and Ψ be a ∂-isometry. Under +gauge transformations on D and D with parameters (z, ζ) and (z, ζ), respectively, Ψ +transforms as +Ψ′ = G−1 ◦ Ψ ◦ G, +(64) +where G is the invertible linear map +G : TH ⊕ F(S) −→ TH ⊕ F(S) +(V, a) �−→ G ((V, a)) := (V + azζ, az) +(65) +and G : TH ⊕ F(S) −→ TH ⊕ F(S) is defined identically but with all the quantities +carrying an underline. As a consequence, the transformation law of the function µ := +A ((n, 0), Ψ(n, 0)) is +µ′ = z−1z−1µ, +(66) +where in order not to overload the notation we denote with the same symbol a function +f ∈ F(∂H) and f ◦ φ−1 ∈ F(∂H). This slight abuse of notation will be used repeatedly +from now on when no confusion arises. +Proof. Let D be hypersurface data. From Def. 2.3 one can write the transformation law +for the tensor A in a compact way as +A′ ((V, a), (W, b)) = A (G(V, a), G(W, b)) +(67) +for all (V, a), (W, b) ∈ TH⊕F(S). To show this we use matrix notation in which vectors +are represented as columns and covectors as rows. The map (65) gets rewritten in matrix +form as +� +V + azζ +az +� += +� +1 +zζ +0 +z +� � +V +a +� +. +Define, therefore +(G) = +� +1 +zζ +0 +z +� +, +(A) = +�γ +ℓT +ℓ +ℓ(2) +� +. +Then (67) can be written in matrix form as +(A′) = (G)T(A)(G). +(68) +To prove this equality (and hence (67)) we compute +(G)T(A)(G) = +� +1 +0 +zζT +z +� �γ +ℓT +ℓ +ℓ(2) +� � +1 +zζ +0 +z +� += +� +γ +ℓT +z (γ(ζ, ·) + ℓ) +z +� +ℓ(ζ) + ℓ(2)� +� � +1 +zζ +0 +z +� += +� +γ +z (γ(ζ, ·) + ℓ)T +z (γ(ζ, ·) + ℓ) +z +� +γ(ζ, ζ) + 2ℓ(ζ) + ℓ(2)� +� +, +which is precisely the matrix form of A′ after taken into account the transformation laws +(8)-(10). Item (2) of Def. 4.1 can be also written in matrix form as (Ψ)T(A)(Ψ) = (A). +19 + +To compute the gauge behaviour of Ψ we impose (Ψ′)T(A′)(Ψ′) = (A′) and use equation +(68), +(Ψ′)T(A′)(Ψ′) = (A′) ⇐⇒ (Ψ′)T(G)T(A)(G)(Ψ′) = (G)T(A)(G) +⇐⇒ +� +(G)T�−1 (Ψ′)T(G)T(A)(G)(Ψ′)(G)−1 = (A), +from where we conclude that Ψ = G ◦ Ψ′ ◦ G−1 and hence Ψ′ = G−1 ◦ Ψ ◦ G, which is +(64). The transformation of µ follows from those of Ψ and A. Indeed, by (67), +µ′ := A′ (Ψ′(n′, 0), (n′, 0)) = A ((G ◦ Ψ′) (n′, 0), G(n′, 0)) , +and using (64) together with G ((n′, 0)) = z−1(n, 0) (and its underlined version), +µ′ = A ((Ψ ◦ G) (n′, 0), G(n′, 0)) += A (Ψ (G(n′, 0)) , G(n′, 0)) += z−1z−1A (Ψ ((n, 0)) , (n, 0)) += z−1z−1µ. +Remark 4.5. The transformation law of a normal pair in Def. 3.5 can be rewritten in +terms of the map G defined in (65) by t′ +i = G−1 (ti). +Next we want to define two null and transverse hypersurfaces from an abstract point +of view, i.e., without seeing them as embedded in any ambient spacetime. In [13] the +notion of double embedded CHD was introduced in order to study how two different +CHD fit together in the same spacetime when we identify their boundaries. However, +using the notion of normal pair and the map Ψ, it is no longer necessary to introduce such +object. Indeed, let D and D be embedded CHD with respective embeddings Φ and Φ in +a spacetime (M, g), and suppose Φ(S) = Φ(S) =: S. Let φ : S −→ S be the induced +diffeomorphism and let Ψ be a ∂-isometry satisfying A (Ψ(n, 0), (n, 0)) = µ ∈ F(S). By +Defs. 2.1 and 2.2, the object A (Ψ(n, 0), (n, 0)) can be thought as the scalar product +g(ν, ν) at S, where ν = Φ⋆n and ν = Φ⋆n. Thus, if one wants to define abstractly two +null and transverse hypersurfaces with the same orientation for ν and ν, the function +A (Ψ(n, 0), (n, 0)) must be everywhere negative when n and n point into the interior of +H and H, respectively. Then by Prop. 4.2 the condition A (Ψ(n, 0), (n, 0)) = µ ̸= 0 +fixes uniquely the map Ψ. Moreover, if we want Φ(S) and Φ(S) to correspond to the +same (codimension two) surface in the ambient spacetime, there are several necessary +conditions that they need to fulfil. Firstly, their induced metrics have to agree. Secondly, +their second fundamental forms and torsion one-forms also need to agree. And finally, the +pullback of the ambient Ricci tensor into the two surfaces must agree too. The “zeroth +order” condition, namely that the induced metrics coincide, can be simply written as +φ⋆h = h. In order to write the “first order” conditions, i.e. the ones of the second +fundamental forms and torsion one-forms, we can employ the tools developed in this +paper. As seen in Section 3, these conditions can be expressed abstractly thanks to the +notion of normal pairs. Indeed, identifying the ambient vectors associated to a basis of +normal pairs {ti} with the ambient vectors associated to the normal pairs {Ψ(ti)}, the +second fundamental forms Kti and the torsions ℸ(ti, tj) of S must agree with those of +S, namely KΨ(ti) and ℸ(Ψ(ti), Ψ(tj)). Finally, the “second order” condition, i.e. the one +involving the ambient Ricci tensor, can be expressed abstractly thanks to the abstract +tensor R defined in (34). Indeed, since Φ⋆ Ric = R and Φ⋆ Ric = R (see (35)), the +pullback of the tensors R and R on S and S must also agree. This whole discussion +motivates the following abstract and fully gauge-covariant definition of double null data. +20 + +Definition 4.6. Let H and H be two manifolds with boundaries i : S −→ H and +i : S −→ H, let φ : S −→ S be a diffeomorphism and µ ∈ F(S) everywhere negative. Let +D = {H, γ, ℓ, ℓ(2), Y} and D = {H, γ, ℓ, ℓ(2), Y} be CHD and restrict n and n to point +towards the interior of H and H, respectively. Let Ψ be the unique ∂-isometry such that +A (Ψ(n, 0), (n, 0)) = µ. Let {ti} be a basis of normal pairs of S. Then we say that the +triple {D, D, µ} is double null data (DND) provided that the following conditions hold at +S +φ⋆h = h, +(69) +Ψ⋆ � +KΨ(ti)� += Kti, +(70) +Ψ⋆ (ℸ(Ψ(ti), Ψ(tj))) = ℸ(ti, tj), +(71) +φ⋆ (i⋆R) = i⋆R. +(72) +In the next remark we show that Def. 4.6 is well-defined, i.e., that the compatibility +conditions are independent of the basis of NPs and also gauge invariant. +Remark 4.7. Conditions φ⋆h = h and φ⋆ (i⋆R) = i⋆R are automatically gauge invari- +ant by virtue of the transformations laws (8) and (36). Then it suffices to show the gauge +invariance of (70)-(71). We start with (70). Let (z, ζ) and (z, ζ) be gauge parameters +and denote by t′ +i the gauge transformed normal pair of ti. Firstly, the RHS of (70) is +simply Kt′ +i as a direct consequence of the gauge invariance of the second fundamental +form in Lemma 3.6. We need to show that the RHS can also be written with all objects +gauge transformed. Let G, G be defined as in Prop. 4.4. By Lemma 3.6 the LHS of (70) +can be written as +Ψ⋆ � +KΨ(ti)� += Ψ⋆ � +KG−1(Ψ(ti))� +, +since the second fundamental form K along a normal pair is gauge invariant and G−1 (Ψ(ti)) +is the gauge transformation of the normal pair Ψ(ti) with gauge parameters (z, ζ) (see +Remark 4.5). Then, using G−1 ◦ Ψ = Ψ′ ◦ G−1 (by Prop. 4.4), +Ψ⋆ � +KG−1(Ψ(ti))� += Ψ⋆ � +KΨ′(G−1(ti))� += Ψ⋆ � +K(Ψ′(t′ +i))� +, +where in the last equality we used again t′ +i = G−1ti. Noting that Ψ ((X, 0)) = Ψ′ ((X, 0)) +for every X ∈ X(S), the LHS of equation (70) finally gets rewritten as +Ψ′⋆ � +K(Ψ′(t′ +i))� +, +which is exactly the LHS of (70) with all quantities gauge transformed. This establishes +the gauge covariance of conditions (70). Moreover, by Lemma 3.4, (70) are also inde- +pendent of the basis of NPs. +The gauge invariance of conditions (71) can be shown in a similar way as in the case +of the second fundamental form. Firstly, the RHS of (71) is simply ℸ(t′ +i, t′ +j), by the +gauge invariance of ℸ in Prop. 3.15. To see that the LHS of (71) can be written with +all quantities gauge transformed, we use again G−1 ◦ Ψ = Ψ′ ◦ G−1, t′ +i = G−1ti and +Ψ ((X, 0)) = Ψ′ ((X, 0)), so that +Ψ⋆ (ℸ(Ψ(ti), Ψ(tj))) = Ψ⋆ � +ℸ(G−1 (Ψ(ti)) , G−1 (Ψ(tj))) +� += Ψ′⋆ � +ℸ(Ψ′(t′ +i), Ψ′(t′ +j)) +� +, +21 + +where in the first equality we invoked again the gauge invariance of ℸ (Prop. 3.15). +Thus, the gauge covariance of (71) is also established. Finally, by Prop. 3.13 under a +change of basis of NPs ˆti = Ωk +i tk, the RHS of (71) transforms as +ℸ(ˆti,ˆtj) = Ωk +i Ωl +jℸ(tk, tl) + Ωk +i MkldΩl +j, +whereas the transformation of the LHS is +Ψ⋆ � +ℸ(Ψ(ˆti), Ψ(ˆtj)) +� += Ψ⋆ �� +Ωk +i ◦ φ−1� � +Ωl +j ◦ φ−1� +ℸ(Ψ(tk), Ψ(tl)) + +� +Ωk +i ◦ φ−1� +M kld +� +Ωl +j ◦ φ−1�� += Ωk +i Ωl +jΨ⋆ (ℸ(Ψ(tk), Ψ(tl))) + Ωk +i MkldΩl +j, +where we used Ψ⋆M kl = Mkl (by item (2) in Def. 4.1), Ψ(ˆti) = Ψ +� +Ωk +i tk +� += +� +Ωk +i ◦ φ−1� +Ψ(tk) +and that the pullback commutes with the exterior derivative. Since both sides in (71) +transform in the same way, conditions (71) are invariant under change of basis of NPs. +Remark 4.8. The compatibility condition (72), namely φ⋆ (i⋆R) = i⋆R, played no +essential role in [13] because in that paper we were interested in data satisfying the +abstract constraint equations +R = +2Λ +m − 1γ +and +R = +2Λ +m − 1γ, +(73) +so (72) was automatically fulfilled. When (73) do not hold (e.g. when the field equations +are not the Einstein vacuum field equations or no equations whatsoever are imposed) +adding this condition is essential. +An explicit case is Theorem 4.15 below where we +prove that given double null data {D, D, µ}, there always exists a spacetime in which +{D, D, µ} can be embedded. It turns out that without the condition φ⋆ (i⋆R) = i⋆R the +result would not be true. +DND has gauge freedom on each component linked by the behaviour of µ as described +in Prop. 4.4. Thus, we put forward the following definition. +Definition 4.9. Let {D, D, µ} be DND and z ∈ F ⋆(H), z ∈ F ⋆(H), ζ ∈ X(H) and +ζ ∈ X(H). The transformed double null data is given by G ({D, D, µ}) := {D′, D′, µ′}, +where D′ and D′ are the transformed CHD in the sense of Definition 2.3 and +µ′ := z−1z−1µ. +(74) +This definition guarantees that when {D, D, µ} is double null data, then G ({D, D, µ}) +is double null data too. This is a straightforward consequence of the gauge covariance of +the compatibility conditions (see Remark 4.7). In order to connect the abstract definition +of double null data with the geometric idea of two null and transverse hypersurfaces, it +is necessary to extend the notion of embeddedness to the context of double null data. +Definition 4.10. Let {D, D, µ} be DND and (M, g) a spacetime. We say that {D, D, µ} +is embedded double null data on (M, g) with riggings ξ, ξ and embeddings Φ, Φ, respec- +tively, provided that +1. D (resp. D) is embedded in (M, g) with embedding Φ (resp. Φ) and rigging ξ +(resp. ξ) in the sense of Def. 2.2 and Φ(S) = Φ(S) =: S. +2. µ(p) = g(ν, ν)|Φ(p) for all p ∈ S, where ν = Φ⋆n and ν = Φ⋆n. +22 + +Definitions 2.2, 4.6 and 4.10 establishes our intended correspondence between embedded +double null data and two transverse, null hypersurfaces. Moreover, since the quantity +A (Ψ(n, 0), (n, 0)) is assumed to be negative when n and n point into the interior of H +and H, respectively, the two hypersurfaces have the same time-orientation. +Let us summarize what we have done so far. In Definition 4.6 we have introduced a +completely abstract object called double null data that captures the idea of two null +and transverse hypersurfaces but without seeing them as embedded in any ambient +spacetime. This object must satisfy certain compatibility conditions at the “corner”, +which are clearly necessary for any DND to be able to be embedded. These conditions +have been written in a fully gauge-covariant way thanks to the notion of normal pair. +They are also independent of the basis of NPs. In the following Remark we connect +the new completely general definition of double null data with the previous one given +in Def. 7.5 of [13], where the compatibility equations were written in a particular gauge +assuming strong conditions at the boundary as well as a very particular basis of normal +pairs. +Remark 4.11. Using Lemma 3.3 and Remark 3.16, the compatibility conditions (70) +can be written in the basis of normal pairs {tn = (n, 0), tθ = (θ, 1)} as +µΨ⋆ (Υ + Θ) = χ, +(75) +µ−1Ψ⋆χ = Υ + Θ. +(76) +In addition, using again Remark 3.16 and Corollary 3.14, (71) can be written as +ℸ(tθ, tn) = (Ψ⋆ℸ) (Ψ(tθ), Ψ(tn)) = (Ψ⋆ℸ) +� +µ−1tn, µtθ +� += Ψ⋆τ + d(log |µ|), +so employing Prop. 3.11 it yields +τ + Ψ⋆τ = −d(log |µ|). +(77) +Given D and D characteristic hypersurface data, there exists a class of gauges in which +ℓ∥ = 0, ℓ(2) = 0 on S and ℓ∥ = 0, ℓ(2) = 0 on S. The existence of these class of gauges +was established in [13, Lemma 7.2] where it was also proved that this family of gauges +was parametrized by pairs (z, ζ) and (z, ζ) satisfying ζ|S = 0 and ζ|S = 0. Particularizing +equations (75), (76) and (77) to this class of gauges, it yields +µY(X, Y ) = K(X, Y ), +(78) +µ−1K(X, Y ) = Y(X, Y ), +(79) +Π(X, n) + Π(X, n) = −X(log |µ|) +(80) +for every X, Y ∈ X(S) (here we omit the pushforward φ⋆ for simplicity). These equations +are the same as the compatibility conditions (122)-(124) in [13]. We conclude that Def. +4.6 generalizes our previous definition of double null data in a fully general gauge and +in any basis of normal pairs. +So far it is clear that the compatibility conditions (69)-(72) are necessary for a double +null data to be embeddable in some spacetime. The rest of this section is devoted to +show that these equations are not only necessary but also sufficient, i.e., that if (69)-(72) +hold, then there exists a spacetime in which the double null data can be embedded. Here +we are not interested in solving any spacetime field equations, we simply want to find +23 + +that there always exists a spacetime where the data can be embedded, with the aim of +showing that we have not forgotten any additional restriction on the data that might +have been necessary. +The construction of the spacetime will be based on the harmonic gauge. In Theorem 2.6 +we have already recalled the result in [13] that guarantees the existence of a harmonic +gauge associated to a set of m functionally independent functions on H. We now choose +functions {xa} = {u, xA} on H and {xa} = {u, xA} on H with the following properties: +(i) +n(u) ̸= 0, u|S = 0, n(xA) = 0 and {xA} is a local coordinate system on S +(ii) n(u) ̸= 0, u|S = 0, n(xA) = 0 and {xA} is a local coordinate system on S +(iii) xA ◦ φ = xA + + + . +(81) +Lemma 2.7 guarantees the existence of a harmonic gauge associated to the functions +{u, xA} (resp. {u, xA}) on H (resp. H) satisfying ℓ∥ = 0, ℓ(2) = 0 on S and ℓ∥ = 0, +ℓ(2) = 0 on S. Moreover, the residual gauge freedom is parametrized by pairs (z, z) ∈ +F ⋆(S) × F ⋆(S). One can exploit this freedom to fix the value of the functions n(u) and +n(u) at S and S, respectively. +Lemma 4.12. Let {D, D, µ} be DND and consider two set of independent functions +{u, xA} and {u, xA} satisfying conditions (81). Then there exists a unique harmonic +gauge w.r.t {u, xA} and {u, xA} in D and D, respectively, in which ℓ∥ = 0, ℓ(2) = 0, +n(u) = µ on S and ℓ∥ = 0, ℓ(2) = 0, n(u) = µ on S. +Proof. The transformation law of the functions n(u) and n(u) follow from that of n +(see (13)), n′(u) = z−1n(u) and n′(u) = z−1n(u). Recalling the transformation of µ +in (66), namely µ′ = z−1z−1µ, one can choose z = µn(u)−1 and z = µn(u)−1 so that +µ′ = n′(u) = n′(u). +Applying Proposition 2.8 it follows that when the DND is written in the unique gauge +defined in the previous Lemma, additional relations between the data at the boundary +appear. +Proposition 4.13. Let {D, D, µ} be DND written in the harmonic gauge of Lemma 4.12 +w.r.t {u, xA} and {u, xA}. Then for every X ∈ X(S) the following relations hold at S +(we omit the pushforward φ⋆ for simplicity) +2Y(n, n) = µn +� +ℓ(2)� +, +(82) +2Y(n, n) = µn +� +ℓ(2)� +, +(83) +2Π(X, n) = (Lnℓ) (X) − X (log |µ|) − (Lnℓ) (X), +(84) +2Π(X, n) = (Lnℓ) (X) − X (log |µ|) − (Lnℓ) (X). +(85) +Proof. Firstly, from equation (39) it follows n +� +ℓ(2)� += trh Υ on S, which together with +(79) and (37) yields (83). Equation (82) is analogous. Secondly, from (40) and xA ◦ φ = +xA, +2 (Lnℓ + Π(·, n)) (gradhxA) = □hxA = □hxA = 2 (Lnℓ + Π(·, n)) (gradhxA). +Employing (80), equations (84) and (85) follow at once. +24 + +In the following remark we compute the compatibility condition (72) in the gauge defined +in Lemma 4.12, for which we first need to write the pullback of the tensor R into a +section. This has been computed in [8] in the case of general null hypersurface data. +Their result is fully diffeomorphism covariant, the dependence on the tensor Y is explicit +and is written without assuming any gauge condition. However, since we only need R +in a very specific gauge and to make this article as self-contained as possible, we redo +this computation in Appendix A assuming a gauge in which both ℓ(2) and ℓ∥ vanish at +the section. +Remark 4.14. In this remark we compute the compatibility condition i⋆R = i⋆R on S +in the gauge defined in Lemma 4.12 (we omit the pullback φ⋆ for simplicity). In Lemma +A.1 of Appendix A we found that the pullback to S of the abstract constraint tensor R +as defined in (34) in a gauge in which ℓ∥ = 0 and ℓ(2) = 0 on S is +RAB = Rh +AB + (2Y(n, n) − trh K) YAB + +� +n +� +ℓ(2)� +− trh Y +� +KAB ++ 4hCDKC(AYB)D + 2∇h +(AτB) + 2∇h +(A (Lnℓ)B) − 2LnΠ(AB) − 2τAτB, +(86) +and analogously the pullback of R into S in the same gauge is +RAB = Rh +AB + (2Y(n, n) − trh K) YAB + +� +n +� +ℓ(2)� +− trh Y +� +KAB ++ 4hCDKC(AYB)D + 2∇h +(Aτ B) + 2∇h +(A (Lnℓ)B) − 2LnΠ(AB) − 2τ Aτ B. +(87) +From condition (69), namely that h +S= h, it follows Rh +AB +S= Rh +AB. From (78)-(79), the +terms trh K YAB, trh Y KAB and 4hCDKC(AYB)D from (86) are equal to the terms +trh Y KAB, trh K YAB and 4hCDKC(AYB)D from (87), respectively. Finally from condi- +tion (77) we can substitute τ in (87) in terms of τ and d log |µ|. Then, the compatibility +condition RAB = RAB in the gauge in which ℓ∥ = ℓ∥ = 0 and ℓ(2) = ℓ(2) = 0 on S and +S can be written as +(LnΠ)(AB) − (LnΠ)(AB) = +� +Y(n, n) − 1 +2µn +� +ℓ(2)�� +YAB + +�1 +2n +� +ℓ(2)� +− µ−1Y(n, n) +� +KAB ++ 2∇h +(AτB) + ∇h +(A (Lnℓ)B) − ∇h +(A (Lnℓ)B) + ∇h +(A∇h +B) log |µ| (88) ++ 2τ(A∇h +B) log |µ| + ∇h +A log |µ|∇h +B log |µ|, +We now restrict further the gauge so that we are in the harmonic gauge of Lemma 4.12. +Taking into account (82)-(85) and the fact that Π(X, n) = τ(X) for every X ∈ X(S), +the first and second lines of the RHS of (88) vanish. Replacing the term 2τA of the third +line by (Lnℓ)A − ∇h +A log |µ| − (Lnℓ)A (see (84)), equation (88) finally reads +(LnΠ)(AB) − (LnΠ)(AB) = (Lnℓ)(A ∇h +B) log |µ| − (Lnℓ)(A ∇h +B) log |µ|. +(89) +We now have the necessary ingredients to show that (69)-(72) is everything one needs +to make sure that double null data can be embedded in some spacetime, i.e., that the +definition is complete and we are not missing any extra conditions. +Theorem 4.15. Let {D, D, µ} be double null data. Then there exists a spacetime (M, g), +embeddings Φ : H ֒→ M, Φ : H ֒→ M and vector fields ξ, ξ along Φ(H) and Φ(H), +respectively, such that {D, D, µ} is embedded double null data in (M, g) with embeddings +Φ, Φ and riggings ξ, ξ, respectively. +25 + +Proof. Let {D, D, µ} be double null data, D = {H, γ, ℓ, ℓ(2), Y} and D = {H, γ, ℓ, ℓ(2), Y}. +Consider a set of independent functions {u, xA} and {u, xA} satisfying the conditions of +Lemma 4.12, namely (81). Henceforth the coordinates u and u will be assumed to be +≥ 0. We write the data in the gauge of Lemma 4.12 with respect to these functions. +Consider the manifold R × R × S and use u and u as the natural coordinates in the +first and second factors, respectively. +We shall work in the manifold with boundary +M := {u, u ≥ 0} ⊂ R2 × S. Any (local) coordinate system {xA} in S extends to a +(local) coordinate system {u, u, xA} in M such that {u, xA} (resp. {u, xA}) restricted to +H (resp. H) are the given coordinates on H (resp. H), as well as u|H = 0 and u|H = 0. +We define the embeddings +Φ : H +−→ M +Φ : H +−→ M +(u, xA) +�−→ (u = 0, u, xA) +(u, xA) +�−→ (u, u = 0, xA). +(90) +Thus, Φ(H) = {u = 0} and Φ(H) = {u = 0}. We denote by S the intersection of Φ(H) +and Φ(H), namely S := {u = u = 0} ⊂ M. Let ξ and ξ be defined by ξ := ∂u and +ξ := ∂u in the coordinate system we have introduced. In these coordinates we also have +n = λ∂u and n = λ∂u, where λ := n(u) and λ := n(u). In order to prove the theorem we +only need to construct a smooth metric g on M inducing the given data on H ∪ H (we +do not write Φ or Φ for simplicity) w.r.t the riggings ξ and ξ, i.e., +gu u|H += 0, +gu u|H += ℓ(2), +gu u|H += ℓ(2), +gu u|H += 0, +gu A|H += 0, +gu A|H += ℓA, +gu A|H += ℓA, +gu A|H += 0, +gAB|H += γAB, +gAB|H += γAB, +gu u|H += λ−1, +gu u|H += λ−1, +(91) +gu u|S = µ−1, +(92) +and +1 +2 (Lξg)ab = Yab, +1 +2 +� +Lξg +� +ab = Yab, +(93) +where ℓA := ℓ(∂xA), γAB := γ(∂xA, ∂xB) and similarly on H. The conditions of the first +line of (91) follow from the fact that γ(n, ·) = 0 and Φ⋆ (g(ξ, ξ)) = ℓ(2) (see (6)). The +second line also follows from γ(n, ·) = 0 as well as from Φ⋆ (g(ξ, ·)) = ℓ. The third line fol- +lows directly from Φ⋆g = γ. The fourth line follows from 1 = g(ξ, ν) = g(∂u, λ∂u) = λgu u +and its underlined version. These four lines constitute the metric part of the data. Con- +dition (92) follows from µ = g(ν, ν)|S = λλgu u and the fact that λ = λ = µ on S. +Finally, conditions (93) guarantee that the metric g also induces the Y tensor (see (7)). +In order to construct such g, our strategy is to extend the components of the hypersurface +data tensors in these coordinates to all M and define the components of the metric g in +such a way that it induces the given data on H ∪ H. Thus, we introduce the notation +f H to denote the extension of the function f ∈ F(H) off H satisfying ξ (f) = 0. We +define f H analogously, i.e., by extending the function f ∈ F(H) by means of ξ(f) = 0. +Moreover, f S will denote the extension of f ∈ F(S) off S satisfying ξ (f) = ξ (f) = 0. +Then we define the components of g in this coordinate system as follows. +• Component gu u: Let gu u be defined on M by gu u := (ℓ(2))H+2(YH +u u−YS +u u)u. Since +YH +u u = YS +u u on H we have gu u|H = ℓ(2). From ℓ(2) S= 0 its extension (ℓ(2))H vanishes +on H and hence gu u|H = 0. Concerning the transverse derivative, we first note +26 + +that for any function f ∈ F(H) it holds ∂u(f H) = (∂uf)H and similarly ∂u(f H) = +(∂uf)H for any function f ∈ F(H). Indeed, ∂u +� +∂u(f H) +� += ∂u +� +∂u(f H) +� += 0 and +∂u +� +(∂uf)H� += 0 by construction, so the function ∂u(f H)−(∂uf)H is constant along +each integral curve of ∂u, and since on H ∂u(f H) = (∂uf)H = ∂uf, we conclude that +∂u(f H) = (∂uf)H (and similarly ∂u(f H) = (∂uf)H on M). Now, condition (82) +together with n = λ∂u, n = λ∂u and λ = λ on S gives ∂uℓ(2) S= 2Yu u. Therefore, +all along H their extensions agree, (∂uℓ(2))H = ∂u +� +(ℓ(2))H� += 2YS +u u. Consequently, +1 +2 (Lξg)u u = 1 +2∂ugu u = 1 +2∂u +� +(ℓ(2))H� ++ YH +u u − YS +u u +H= Yu u. +• Component gu u: Analogously, we define gu u := (ℓ(2))H + 2 +� +YH +u u − YS +u u +� +u. By +symmetry of the construction this also induces the given data on H and H. +• Component gu u: Let gu u be defined on M by gu u := (λ−1)H + +� +λ−1�H − (µ−1)S. +Since µ = λ = λ on S, gu u|H = λ−1 and gu u|H = λ−1 (and they match on S and +fulfil condition (92)). +• Components gu A: Let gu A := ℓH +A + 2(YH +u A − YS +u A)u on M. Since YH +u A = YS +u A +on H we have gu A|H = ℓA. From ℓA = 0 on S, its extension ℓH +A also vanishes on +H, and then gu A|H = 0. In order to see that these gu A induce the corresponding +components of the Y tensor we start by writing equation (84) in the coordinate +system {u, u, xA}, i.e., taking X = ∂xA, n = λ∂u and n = λ∂u. Using µ +S= λ, +2λΠA u +S= λ∂uℓA + ℓ(∂u)∂xA(λ) − ∂xA log |λ| − λ∂uℓA − ℓ(∂u)∂xA(λ) +S= λ∂uℓA + λ−1∂xA(λ) − ∂xA log |λ| − λ∂uℓA − λ−1∂xA(λ) +S= λ∂uℓA − ∂xA log |λ| − λ∂uℓA, +(94) +where in the first line we used the well-known formula LfXω = fLXω + ω(X)df +valid for any function f, vector X and one-form ω, in the second line we used +ℓ(n) = λℓ(∂u) = 1 and thus ℓ(∂u) = λ−1 (and its underlined version) and in the +third line λ∂xA(λ) +S= λ∂xA(λ) since λ = λ on S. The value of ΠA u can be computed +from (23) and (17), namely +2ΠA u = 2YA u + 2FA u = 2YA u + ∂xAλ−1 − ∂uℓA, +(95) +where again we used ℓu = λ−1. Inserting (95) evaluated at S into (94) and taking +again into account that λ = λ on S, it yields 2Yu A = ∂uℓA on S and therefore +2YS +u A = (∂uℓA)H = ∂u +� +ℓH +A +� +on H. Hence, +1 +2 (Lξg)u A = 1 +2∂ugu A = 1 +2∂u +� +ℓH +A +� ++ YH +u A − YS +u A +H= Yu A. +• Components gu A: Analogously, we define gu A := ℓH +A + 2(YH +u A − YS +u A)u on M, +which by symmetry also induces the given data on H and H. +• Components gAB: Let h be the induced metric on S. We define the functions gAB +on M by means of +gAB := γH +AB + γH +AB − hS +AB + 2 +� +YH +AB − YS +AB +� +u + 2 +� +YH +AB − YS +AB +� +u − 2(∂uYAB)Suu. +27 + +Since on H γH +AB = hS +AB and YH +AB = YS +AB, and on H γH +AB = hS +AB and YH +AB = YS +AB, +we have gAB|H = γAB and gAB|H = γAB. Moreover, from (18) and n = λ∂u, we +have ∂uγAB = 2λ−1KAB on H, so in particular ∂uγAB = 2λ−1KAB on S. Using +λ = µ and µ−1KAB = YAB on S (see (79)) it follows that (∂uγAB)H = ∂u +� +γH +AB +� += +2YS +AB on H, and thus +1 +2 (Lξg)AB = 1 +2∂ugAB += 1 +2∂u(γH +AB) + YH +AB − YS +AB + ∂u(YH +AB)u − (∂uYAB)S u +H= 1 +2∂u(γAB)H + YH +AB − YS +AB +H= YAB, +where in the third equality we used ∂u +� +YH +AB +� += (∂uYAB)S on H. Before computing +Lξg on H we need to write equation (89) in the coordinate system {u, u, xA}. Since +[n, ∂xA] = [λ∂u, ∂xA] = −∂xAλ ∂u and λ = µ on S, +(LnΠ)AB +S= n (ΠAB) + ∂xA(log |µ|)Π (n, ∂xB) + ∂xB(log |µ|)Π (∂xA, n) . +Using (24) and the fact that F is antisymmetric, +(LnΠ)AB + (LnΠ)BA +S= 2n (YAB) + (2Π(∂xB, n) + (Lnℓ) (∂xB)) ∂xA(log |µ|) ++ (2Π(∂xA, n) + (Lnℓ) (∂xA)) ∂xB(log |µ|) +S= 2λ∂u (YAB) + (Lnℓ) (∂xB)∂xA(log |µ|) + (Lnℓ) (∂xA)∂xB(log |µ|) +− 2∂xA(log |µ|)∂xB(log |µ|), +where in the second equality we used (84). Then, equation (89) in this coordinate +system becomes simply +∂u (YAB) +S= ∂u (YAB) , +(96) +and then the quantity Lξg on H is finally given by +1 +2 +� +Lξg +� +AB = 1 +2∂ugAB += 1 +2∂u +� +γH +AB +� ++ ∂u +� +YH +AB +� +u + YH +AB − YS +AB − (∂uYAB)S u +H= 1 +2∂u (γAB)H + YH +AB − YS +AB +H= YAB, +where in the third line we used that on H ∂u +� +YH +AB +� += (∂uYAB)S = (∂uYAB)S +(the second equality following from (96)), and in the fourth line that (∂uγAB)H = +∂u +� +γH +AB +� += 2YS +AB on H. +Given that gµν fulfils all conditions (91)-(93), we conclude that {D, D, µ} is embedded +double null data in (M, g) with embeddings Φ, Φ and riggings ξ, ξ, respectively. +28 + +The previous Theorem shows that any double null data can be embedded in some space- +time. +It then arises the natural question of whether it can be also embedded in a +spacetime solution of the Einstein equations. By (35), if {D, D, µ} is embedded DND on +an (m + 1)-dimensional spacetime (M, g) solution of the Λ-vacuum equations, namely +Ric = +2Λ +m − 1g, +then it must satisfy +R = +2Λ +m − 1γ +and +R = +2Λ +m − 1γ, +where R is the abstract tensor defined in (34) (and analogously for R). Thus, these +restrictions are necessary for {D, D, µ} to be embedded in a spacetime solution of the +Λ-vacuum Einstein equations. The main result in [13] proves that they are also sufficient, +as we summarize next. +Definition 4.16. Let {D, D, µ} be double null data of dimension m > 1. +We say +that a Lorentzian manifold (M, g) is a development of {D, D, µ} provided there exist +embeddings Φ, Φ and riggings ξ, ξ such that {D, D, µ} is embedded DND in (M, g) with +embeddings Φ, Φ and riggings ξ, ξ in the sense of Def. 4.10 and Φ(H) ∪ Φ(H) = ∂M. +With this definition we can restate Theorem 7.15 of [13] in the following way. +Theorem 4.17. Let {D, D, µ} be double null data of dimension m > 1 as defined in +Def. 4.6 satisfying the abstract constraint equations +R = +2Λ +m − 1γ +and +R = +2Λ +m − 1γ, +(97) +where R is defined in (34), R is its underlined version and Λ ∈ R. Then there ex- +ists a development (M, g) of {D, D, µ} (possibly restricted if necessary) solution of the +Λ-vacuum Einstein equations. Moreover, for any two such developments (M, g) and +( � +M, �g), there exist neighbourhoods of H ∪ H, U ⊆ M and �U ⊆ � +M, and a diffeomor- +phism ϕ : U −→ �U such that ϕ⋆�g = g. +Remark 4.18. Theorems 4.15 and 4.17 establish a very clear hierarchy between the +compatibility conditions and the constraint equations. The former are the necessary and +sufficient conditions for a DND to be able to be embedded in some spacetime, whereas the +later are necessary and sufficient for the DND to be embedded in a spacetime solution of +the Einstein field equations. +5 +Isometry between Double Null Data +Given two double null data, there arises the natural question of under which conditions +their developments are the same (up to isometry). In this section we establish the neces- +sary and sufficient conditions for two double null data to define two isometric spacetimes. +We start with a definition to fix some notation. +Definition 5.1. Let D = {H, γ, ℓ, ℓ(2), Y} be hypersurface data and ψ : � +H −→ H a +diffeomorphism. We define the pull-back hypersurface data ψ⋆D by +ψ⋆D := +� +� +H, �γ := ψ⋆γ, �ℓ := ψ⋆ℓ, �ℓ(2) := ψ⋆ℓ(2), �Y := ψ⋆Y +� +. +29 + +From Def. +2.1 and the fact that ψ is a diffeomorphism, it follows that ψ⋆D is still +hypersurface data. Moreover, from (1)-(5) having a unique solution for {P, n, n(2)} given +{γ, ℓ, ℓ(2)}, it follows that �P = ψ⋆P, �n = ψ⋆n and �n(2) = ψ⋆n(2). Thus, the causal +character of D is the same as the one of ψ⋆D, and in particular if D is null, so it is ψ⋆D. +The previous definition can be extended to the context of double null data as follows. +Definition 5.2. Let {D, D, µ} be double null data and ψ : � +H −→ H, ψ : � +H −→ H +diffeomorphisms. The pull-back double null data Ξ⋆{D, D, µ} is defined as +Ξ⋆{D, D, µ} := +� +ψ⋆D, ψ⋆D, ψ|⋆ +S(µ) +� +, +where ψ⋆D and ψ⋆D are the pull-backs in the sense of Def. 5.1. +Since ψ and ψ are diffeomorphisms, they preserve the boundaries S and S, and thus the +map �φ := ψ◦φ◦ψ−1 : �S −→ �S is a diffeomorphism. Then, Ξ⋆{D, D, µ} is still double null +data, since it satisfies Def. 4.6 with �φ and �µ := ψ|⋆ +S(µ). In the following proposition we +find the necessary conditions for two DND to define two isometric Lorentzian manifolds. +Proposition 5.3. Let {D, D, µ} and { �D, �D, �µ} be double null data satisfying the con- +straint equations (97) and let (M, g) and ( � +M, �g) be respective developments. Suppose +that there exists an isometry ϕ : M −→ � +M. Then there exist gauge parameters (z, ζ) +and (z, ζ) in D and D, respectively, and a map Ξ⋆ as in Def. 5.2 such that +Ξ⋆{ �D, �D, �µ} = G ({D, D, µ}) . +Proof. Let �xµ = {�u, �u, �xA} be coordinates on � +M whose restrictions on H and H satisfy +(81). Define the coordinates xµ on M by xµ := �xµ ◦ ϕ. Let Φ, Φ and ξ, ξ the embeddings +and the riggings of {D, D, µ} in (M, g), and �Φ, �Φ and �ξ, �ξ be the embeddings and the +riggings of { �D, �D, �µ} in ( � +M, �g). Since (ϕ⋆�ξ)(u) = �ξ(u ◦ ϕ−1) = �ξ(�u) ̸= 0, there exist +z ∈ F ⋆(H) and ζ ∈ X(H) such that ϕ⋆�ξ = z(ξ + Φ⋆ζ) along Φ(H). Since Φ(H) is diffeo- +morphic to �Φ( � +H) via ϕ and both Φ and �Φ are embeddings, there exists a diffeomorphism +ψ making the following diagram commutative +H +� +H +M +� +M +ψ +Φ +�Φ +ϕ +Then +ψ⋆�γ = ψ⋆�Φ⋆�g = Φ⋆ϕ⋆�g = Φ⋆g = γ, +and +ψ⋆�ℓ = ψ⋆�Φ⋆� +�g(�ξ, ·) +� += Φ⋆ϕ⋆� +�g(�ξ, ·) +� += Φ⋆ (g(z(ξ + Φ⋆ζ), ·)) = z(ℓ + γ(ζ, ·)). +Concerning �ℓ(2), +ψ⋆�ℓ(2) = ψ⋆�Φ⋆� +�g(�ξ, �ξ) +� += Φ⋆ � +z2g(ξ, ξ) + 2z2g(ξ, Φ⋆ζ) + z2g(Φ⋆ζ, Φ⋆ζ) +� += z2(ℓ(2) + 2ℓ(ζ) + γ(ζ, ζ)). +30 + +Finally, +ψ⋆ �Y = 1 +2ψ⋆�Φ⋆L�ξ�g = 1 +2Φ⋆ϕ⋆L�ξ�g = 1 +2Φ⋆ � +Lz(ξ+Φ⋆ζ)g +� += zY + dz ⊗s ℓ + 1 +2Lzζγ, +where the third equality holds because ϕ⋆�ξ = z (ξ + Φ⋆ζ), ϕ⋆�g = g and3 ϕ⋆� +L�ξ�g +� += +Lϕ⋆�ξ +� +ϕ⋆g +� +. Thus, recalling Def. 2.2, ψ⋆ �D = G(z,ζ)(D). The same argument on H proves +that there exist gauge parameters (z, ζ) on D such that ψ⋆ �D = G(z,ζ)(D). Finally, taking +into account item 2. of Def 4.10, +ψ|⋆ +S(�µ) = Φ|⋆ +Sϕ⋆ (�g(�ν, �ν)) = Φ|⋆ +S +� +g(z−1ν, z−1ν) +� += z−1z−1µ. +Comparing with (66), the result follows. +The previous proposition motivates defining the notion of isometric double null data as +follows. +Definition 5.4. We say that two double null data {D, D, µ} and { �D, �D, �µ} are isometric +if there exist diffeomorphisms ψ : H −→ � +H and ψ : H −→ �H and gauge parameters (z, ζ) +and (z, ζ) in D and D, respectively, such that the pull-back double null data Ξ⋆{ �D, �D, �µ} +satisfies +Ξ⋆{ �D, �D, �µ} = G ({D, D, µ}) . +We conclude this paper by proving that the necessary conditions of Prop. 5.3 are also suf- +ficient. This result gives a geometric uniqueness statement of the characteristic problem +of the Einstein field equations. Indeed, two isometric initial data are indistinguishable +from a geometric point of view and thus they should have “the same” developments. +The precise statement of the notion of uniqueness is given in the following Theorem. +Theorem 5.5. Let {D, D, µ}, { �D, �D, �µ} be two isometric DND in the sense of Def. 5.4 +satisfying the abstract constraint equations (97), and let (M, g), ( � +M, �g) be respective +developments. Then there exist neighbourhoods U ⊆ M and �U ⊆ � +M of H ∪ H and +� +H ∪ � +H, respectively, and a diffeomorphism ϕ : U −→ �U such that ϕ⋆�g = g. +Proof. We start by writing { �D, �D, �µ} in the gauge of Lemma 4.12 w.r.t some coordi- +nates {�xa} = {�u, �xA} and {�xa} = {�u, �xA} in H and H satisfying (81), and {D, D, µ} in +the gauge in which Ξ⋆{ �D, �D, �µ} = {D, D, µ} holds. We want to show that Ξ⋆{ �D, �D, �µ} +is written in the gauge of Lemma 4.12 w.r.t the coordinates {xa} := {�xa ◦ ψ} and +{xa} := {�xa ◦ ψ}. Let �V be the vector field defined in Thm. 2.6 w.r.t {�xa}. First +we prove that ψ⋆ �V is again the vector of Theorem 2.6 but w.r.t {ψ⋆�xa} (and anal- +ogously on H). +Let {�ec} be a (local) basis of X( � +H) and {ec := ψ⋆�ec} a (local) ba- +sis of X(H). +Since ψ⋆ (�ec(�xa)) = ec(xa), it follows ψ⋆ �Bca = Bca. +Moreover, since +ψ⋆ �∇ = ∇ (because Ξ⋆{ �D, �D, �µ} = {D, D, µ}), the pull-back of the Hessian of a function +is the Hessian of the pullback of that function, and since ψ⋆ �P = P, it turns out that +ψ⋆� �P ab �∇a �∇b�xa� += P ab∇a∇bxa, and thus ψ⋆ �V c = ψ⋆� �Bca �□ � +P �xa� += Bca□Pxa. There- +fore Ξ⋆{ �D, �D, �µ} is written in a harmonic gauge w.r.t {xa} and {xa}. Moreover, since +3This follows from the known formula ϕ⋆ (Lϕ⋆XT ) = LX (ϕ⋆T ) valid for any diffeomorphism ϕ, +vector X and tensor T particularized to T = �g and ϕ⋆X = �ξ =⇒ X = ϕ⋆�ξ = z (ξ + Φ⋆ζ). +31 + +PSfrag replacements +(M, g) +( � +M, �g) +ϕ +U +�U +Figure 1: Given isometric DND, there exist isometric neighbourhoods U and �U of the +initial data. +ψ⋆�ℓ(2) = ℓ(2) = 0, ψ⋆�ℓ∥ = ℓ∥ = 0 on S, ψ⋆�ℓ(2) = ℓ(2) = 0, ψ⋆�ℓ∥ = ℓ∥ = 0 on S, as well +as ψ⋆ +S(�µ) = ψ⋆ +S (�n(�u)) = +� +ψ⋆�n +� +(u) on S and similarly ψ|⋆ +S(�µ) = (ψ⋆�n) (u) on S (we omit +the φ⋆ for simplicity), we conclude that the data Ξ⋆{ �D, �D, �µ} is written in the gauge of +Lemma 4.12 w.r.t the coordinates {xa} and {xa}. +Let (M, g) be a development of {D, D, µ} with embeddings Φ, Φ and riggings ξ, ξ and let +( � +M, �g) be the same but everything with a “�”. Let {xµ} be the harmonic coordinates +on (M, g) restricting to the given ones at H ∪ H and satisfying Φ(H) = {u = 0}, +Φ(H) = {u = 0} (and the same with “�”). It is straightforward to see that the rigging +vectors are given by ξ = ∂u, ξ = ∂u and �ξ = ∂�u, �ξ = ∂�u (we refer the reader to the proof +of Theorem 7.15 of [13] for the details). Choosing suitable neighbourhoods U ⊆ M of +H ∪ H and �U ⊆ � +M of � +H ∪ �H (see figure 1), we can define the diffeomorphism ϕ by +xµ = �xµ ◦ ϕ, which by construction restricts to the given diffeomorphisms ψ : H −→ � +H +and ψ : H −→ �H. Since ( � +M, �g) is a solution of the EFE with cosmological constant Λ, +so it is (M, ϕ⋆�g). Moreover, □ϕ⋆�gxµ = 0, so both g and ϕ⋆�g are a solution of the reduced +equations in the coordinates {xµ}. By theorem 1 of [15], in order to prove that g = ϕ⋆�g +on U, we only need to show that their restrictions to H ∪ H agree, which follows at once +after recalling Ξ⋆{ �D, �D, �µ} = {D, D, µ} and using that ϕ restricts to ψ and ψ on H and +H, respectively. +A +Abstract constraint tensor in a section +Let D = {H, γ, ℓ, ℓ(2), Y} be null hypersurface data admitting a section i : S ֒→ H. As +shown in [13, Cor. 4.4], one can always choose a gauge in which ℓ∥ = 0 and ℓ(2) = 0 +on S. In this appendix we compute the pullback of the abstract constraint tensor R as +defined in (34) into S in the aforementioned gauge. +Lemma A.1. Let D = {H, γ, ℓ, ℓ(2), Y} be null hypersurface data admitting a section +i : S ֒→ H and let R be the abstract constraint tensor (34). Then the pullback of R into +S in a gauge in which ℓ(2) = 0 and ℓ∥ = 0 on S takes the following form +RAB = Rh +AB + (2Y(n, n) − trh K) YAB + +� +n +� +ℓ(2)� +− trh Y +� +KAB ++ 4hCDKC(AYB)D + 2∇h +(AτB) + 2∇h +(A (Lnℓ)B) − 2 (LnΠ)(AB) − 2τAτB. +(98) +32 + +Proof. Let A and B be the tensors defined by +Abca := ℓdR +d +bca + 2ℓ(2)∇[aKc]b + Kb[c∇a]ℓ(2), +(99) +Babcd := γafR +f +bcd + 2∇[d +� +Kc]bℓa +� ++ 2ℓ(2)Kb[cKd]a. +(100) +Comparing (34) with (99)-(100), the tensor R can be written as +Rab = BacbdP cd + (Abca + Aacb) nc. +(101) +Let {eA} be a (local) basis of TS with dual basis {θA}, i.e., θA(eB) = δA +B. Then {n, eA} +is a (local) basis of TS ⊕ span{n}, and since ℓ∥ = 0, {ℓ, θA} is the dual basis of {n, eA}. +By equations (3)-(4), the tensor P takes the following form in the basis {n, eA} +P = hABeA ⊗ eB. +(102) +In order to write down the tensor RAB := Rabea +Aeb +B we divide the computation into two +parts, namely BacbdP cdea +Aeb +B and Abcancea +Aeb +B. To calculate the former we need to recall +some results concerning the relation between ∇ and the Levi-Civita connection ∇h of +h := i⋆γ. As a consequence of Prop. 3.5 and equation (30) of [13] together with ℓ∥ = 0 +and ℓ(2) = 0, the relation between ∇ and ∇h is +∇XY = ∇ +h +XY − Y(X, Y )n, +X, Y ∈ X(S). +(103) +As usual, this decomposition allows us to relate the completely tangential components +of the curvature tensor of ∇ with the curvature tensor of ∇h via a Gauss-type identity. +The explicit expression is obtained in [13, Prop. 3.7] and in the present gauge reads +γ +� +W, R(X, Y )Z +� += h +� +W, Rh(X, Y )Z +� +− Y(Y, Z)K(X, W) + Y(X, Z)K(Y, W), (104) +where R and Rh are the curvature tensors of ∇ and ∇h, respectively. We now have all +the ingredients needed to compute BacbdP cdea +Aeb +B. Taking into account that ℓ(2) = 0 and +using (102), +BacbdP cdea +Aeb +B = +� +γcfR +f +adb + 2∇[b +� +Kd]aℓc +�� +hCDec +Ced +Dea +Aeb +B. +(105) +For the first term we employ Gauss identity (104) and equation (102), +γcfR +f +adbhCDec +Ced +Dea +Aeb +B = γ +� +eC, R(eD, eB)eA +� +hCD += Rh +AB − YAB trh K + hCDYDAKBC, +where Rh +AB is the Ricci tensor of h. In this appendix, when no confusion arises we denote +with the same symbol a tensor on H and its pullback on S. For the second term of (105) +we use (21), which in this gauge is ∇aℓb +S= Πab, and the fact that ΠAB +S= YAB (because +ℓ∥ = 0 and thus 2i⋆F = i⋆dℓ = dℓ∥ = 0). Then, +2hCDec +Ced +Dea +Aeb +B∇[b +� +Kd]aℓc +� += 2hCDec +Ced +Dea +Aeb +BKa[d∇b]ℓc = −2hCDKA[BYD]C, +where the first equality holds because ℓ∥ = 0. Combining the two terms, (105) finally +reads +BacbdhCDec +Ced +Dea +Aeb +B = Rh +AB − YAB trh K − KAB trh Y + 2hCDYC(AKB)D. +(106) +33 + +Next we compute the terms of the form Abcancea +Aeb +B. In the present gauge the quantity +Abcanc is simply (note that ℓ(2) is not assumed to be zero off S) +nc � +ℓdR +d +bca + Kb[c∇a]ℓ(2)� += ncℓdR +d +bca − 1 +2Kban +� +ℓ(2)� +, +(107) +where K(n, ·) = 0 has been used. The first term in (107) can be computed directly from +the Ricci identity, +ncℓdR +d +bca = 2nc∇[a∇c]ℓb += 2nc � +∇[aΠc]b − Kb[c∇a]ℓ(2)� += nc∇aΠcb − ∇nΠab + Kabn +� +ℓ(2)� +, +(108) +where we used ∇aℓb = Πab − ℓ(2)Kab and that K(n, ·) = 0. Contracting the first term in +(108) with ea +Aeb +B and using (27) and (24) +ea +Aeb +Bnc∇aΠcb = ea +Aeb +B∇a (Πcbnc) − ea +Aeb +BΠcb∇anc += ea +Aeb +B∇a (Πbcnc + Lnℓb) − ea +Aeb +BΠcb +� +P cdKda − Πadncnd� += eA (Π(eB, n) + (Lnℓ)(eB)) − (Π(·, n) + Lnℓ) +� +∇eAeb +B +� +− hCDec +Ced +Dea +Aeb +BΠcbKda + ea +Aeb +B (Πbc + Lnℓb) Πadncnd. +Introducing (103) and recalling Π(n, n) = Y(n, n), ΠAB = YAB and the fact that in +this gauge Π(eA, n) = τA (see Def. 2.5) this expression finally yields +ea +Aeb +Bnc∇aΠcb = +� +∇h +eA (τ + Lnℓ) +� +(eB) + Y(n, n)YAB +− hCDYCBKDA + τAτB + τA (Lnℓ)B , +(109) +where we also used (Lnℓ) (n) = Ln (ℓ(n)) = 0. Equation (27) can be rewritten as +∇Xn = K♯(X) − Π(X, n)n, +(110) +where K♯ is the endomorphism defined by K♯(X) := P (K(X, ·), ·), or in abstract in- +dex notation (K♯)ab = P acKcb. Using ∇nX = ∇Xn + LnX together with (110), the +contraction of the second term in (108) with tangential directions can be written as +XaY b∇nΠab = n (Π(X, Y )) − Π +� +∇nX, Y +� +− Π +� +X, ∇nY +� += Ln (Π(X, Y )) − Π +� +∇Xn + LnX, Y +� +− Π +� +X, ∇Y n + LnY +� += (LnΠ) (X, Y ) − Π +� +K♯(X) − Π(X, n)n, Y +� +− Π +� +X, K♯(Y ) − Π(Y, n)n +� +, +and therefore +ea +Aeb +B∇nΠab = (LnΠ)AB − hCDYCBKDA + 2τAτB + τA (Lnℓ)B − hCDYACKDB. (111) +Contracting (108) with ea +Aeb +B and introducing (109) and (111), +ea +Aeb +BncℓdR +d +bca = +� +∇h +eA (τ + Lnℓ) +� +(eB) − (LnΠ)AB + Y(n, n)YAB ++ hCDYACKDB − τAτB + n +� +ℓ(2)� +KAB, +and thus +(Aacb + Abca) ncea +Aeb +B = 2∇h +(AτB) + 2∇h +(A (Lnℓ)B) − 2 (LnΠ)(AB) + 2Y(n, n)YAB ++ 2hCDKC(AYB)D − 2τAτB + n +� +ℓ(2)� +KAB, +(112) +Finally, combining (106) and (112), (98) follows. +34 + +Acknowledgements +This work has been supported by Projects PID2021-122938NB-I00 (Spanish Ministe- +rio de Ciencia e Innovaci´on and FEDER “A way of making Europe”) and SA096P20 +(JCyL). G. S´anchez-P´erez also acknowledges support of the PhD. grant FPU20/03751 +from Spanish Ministerio de Universidades. We are very grateful to Miguel Manzano for +useful comments. +References +[1] Cabet, A., Chru´sciel, P. T., and Wafo, R. T. On the characteristic ini- +tial value problem for nonlinear symmetric hyperbolic systems, including Einstein +equations. Dissertationes Mathematicae 515 (2016), 1–67. +[2] Caciotta, G., and Nicol`o, F. Global characteristic problem for Einstein vac- +uum equations with small initial data: (i) the initial data constraints. Journal of +Hyperbolic Differential Equations 2 (2005), 201–277. +[3] Choquet-Bruhat, Y., Chru´sciel, P. T., and Mart´ın-Garc´ıa, J. M. The +Cauchy problem on a characteristic cone for the Einstein equations in arbitrary +dimensions. In Annales Henri Poincar´e (2011), vol. 12, Springer, pp. 419–482. +[4] Chru´sciel, P. T., and Paetz, T.-T. +The many ways of the characteristic +Cauchy problem. Classical and Quantum Gravity 29 (2012). +[5] Hilditch, D., Valiente Kroon, J. A., and Zhao, P. Revisiting the char- +acteristic initial value problem for the vacuum Einstein field equations. General +Relativity and Gravitation 52 (2020), 1–76. +[6] Klainerman, S., and Nicol`o, F. The evolution problem in General Relativity, +vol. 25. Springer Science & Business Media, 2012. +[7] Luk, J. On the local existence for the characteristic initial value problem in General +Relativity. International Mathematics Research Notices 2012 (2012), 4625–4678. +[8] Manzano, M., and Mars, M. In preparation. +[9] Manzano, M., and Mars, M. General matching across Killing horizons of zero +order. arXiv preprint arXiv:2205.08831 (2022). +[10] Mars, M. Constraint equations for general hypersurfaces and applications to shells. +General Relativity and Gravitation 45 (2013), 2175–2221. +[11] Mars, M. Hypersurface data: general properties and Birkhoff theorem in spherical +symmetry. Mediterranean Journal of Mathematics 17 (2020), 1–45. +[12] Mars, M., and Senovilla, J. M. M. Geometry of general hypersurfaces in +spacetime: junction conditions. Classical and Quantum Gravity 10 (1993), 1865. +[13] Mars, M., and S´anchez-P´erez, G. Double Null Data and the Characteristic +Problem in General Relativity. 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University of Chicago Press, 2010. +36 + diff --git a/WtE0T4oBgHgl3EQf3QI4/content/tmp_files/load_file.txt b/WtE0T4oBgHgl3EQf3QI4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c9a3db2827db5f8149f79ab59f4dbc2080592a7b --- /dev/null +++ b/WtE0T4oBgHgl3EQf3QI4/content/tmp_files/load_file.txt @@ -0,0 +1,1074 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf,len=1073 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='02722v1 [gr-qc] 6 Jan 2023 Covariant definition of Double Null Data and geometric uniqueness of the characteristic initial value problem Marc Mars∗ and Gabriel S´anchez-P´erez† Departamento de F´ısica Fundamental,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Universidad de Salamanca Plaza de la Merced s/n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 37008 Salamanca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Spain January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 2023 Abstract The characteristic Cauchy problem of the Einstein field equations has been recently addressed from a completely abstract viewpoint by means of hypersurface data and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' in particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' via the notion of double null data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' However, this definition was given in a partially gauge-fixed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In this paper we generalize the notion of double null data in a fully diffeomorphism and gauge covariant way, and show that the definition is complete by proving that no extra conditions are needed to embed the double null data in some spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The second aim of the paper is to show that the characteristic Cauchy problem satisfies a geometric uniqueness property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Specifically, we introduce a natural notion of isometry at the abstract level such that two double null data that are isometric in this sense give rise to isometric spacetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 1 Introduction The aim of this paper is two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Firstly, we prove a geometric uniqueness result of the Characteristic Cauchy problem in General Relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' A prerequisite for a result of this type is to have an abstract notion of the initial data completely detached from the spacetime one wishes to construct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In the standard Cauchy problem this consists of a Riemannian manifold endowed with a symmetric (0,2)-tensor satisfying the vacuum constraint equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In the null case, not such abstract formulation was known until recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In [13] we have developed a fully detached notion of initial data for the char- acteristic problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The key notions to achieve this are the hypersurface data formalism, the concept of double null data and the so-called constraint tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' However, the defini- tion of double null data in [13] was given in a (partially) gauge-fixed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The second objective of this paper is to address this problem and give a fully gauge-covariant notion of double null data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The hypersurface data formalism was developed in [11, 10] to study general hypersur- faces at a purely abstract level, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=', without reference to any ambient space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' It has been employed recently in the problem of matching of spacetimes across null hypersurfaces [9] and to study the characteristic problem in General Relativity [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' A hypersurface ∗marc@usal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='es †gasape21@usal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='es 1 data set D consists of an abstract manifold H and a set of tensors defined on H, with which one can reconstruct the full ambient metric on the hypersurface and the transverse derivative of its tangential components whenever the data happens to be embedded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Hy- persurface data has an internal gauge structure associated to the freedom in the choice of an everywhere transverse vector (so-called rigging) to the hypersurface in the embed- ded case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In [13] we have particularized the definition of hypersurface data to the null case, so that it describes an abstract null hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' It turns out that that when the data is embedded, the pullback of the ambient Ricci tensor into the null hypersur- face can be written completely in terms of the abstract data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' This allowed us to define and study the characteristic problem for the Einstein equations in a completely abstract way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The data corresponds to two null hypersurfaces intersecting transversely in a space- time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' When the full metric is prescribed in two such null hypersurfaces, Randall [15] has shown that the reduced Einstein equations are well-posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Moreover, if the metric components are prescribed suitably, the ambient spacetime is not only a solution of the reduced equations, but also a solution of the vacuum Einstein equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' This spacetime exists in a neighbourhood of the intersection and the result is valid on any dimension and for all topologies of the intersection submanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' When the intersecting surface is a 2-sphere, Luk shows in [7] that the spacetime can be extended to a neighbourhood of the two initial null hypersurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In dimension four the existence of the solution in a full neighbourhood of the initial data hypersurfaces has been proved in [1] for a large class of symmetric hyperbolic systems (including Einstein equations) irrespectively of the topology of the intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Other approaches to the characteristic problem can be found in [2, 4, 5, 6, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In [3] this Cauchy problem is studied on the future null cone of a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' As already mentioned, in [13] we approached the characteristic initial value problem in a fully abstract way by means of the hypersurface data formalism, and in particular with a new geometric object called double null data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Our formulation detaches the initial data in the characteristic problem from the spacetime and puts the characteristic problem on the same footing as the standard Cauchy problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' A double null data set consists of two null hypersurface data D and D with respective boundaries S and S, together with a non vanishing function at the “corner”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In order for the double null data to be embedded in some spacetime, certain compatibility conditions at S must be fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Geometrically, they arise from the fact that in the embedded case the boundaries are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Con- cretely, these conditions correspond to (i) the induced metrics, (ii) the corresponding torsion one-forms and second fundamental forms, and (iii) the pullbacks of the ambient Ricci tensor be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' As we show in [13], these conditions can be written solely in terms of the abstract data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' While (i) and (iii) was already written gauge-covariantly, the strategy we followed to obtain (ii) was to work in a very particular gauge so that the conditions took a simplified form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Although the existence of such gauge is always granted, defining double null data in a gauge-fixed form is not completely satisfactory from a mathematical and geometric point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' There may be situations where writ- ing the data in some other gauge could simplify the problem, so giving a gauge-covariant definition of double null data becomes necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In this paper we extend the definition of double null data and make it fully gauge covari- ant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The construction is based on the properties of (codimension one) non-degenerate submanifolds of any given null hypersurface data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' For such submanifolds we introduce the notion of normal pair, which abstractly captures the idea of a normal vector to the 2 submanifold when embedded in an ambient space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' It turns out that the (ambient) sec- ond fundamental form of the submanifold along a normal vector can be written solely in terms of the abstract data and the associated normal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' This allows us to define, at the abstract level, the second fundamental form of a submanifold along a normal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In a similar way, one can define abstractly the notion of torsion one-forms associated to a basis of normal pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' It turns out that both the second fundamental form and the torsion one-forms are gauge-covariant, so it is natural to write the compatibility condi- tions for (ii) in terms of these objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' To achieve this one needs to “glue” abstractly the two boundaries together in a (metric) hypersurface data sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' This is accomplished by means of the notion of ∂-isometry, which depends on a diffeomorphism between the boundaries as well as on a non-vanishing function that geometrically corresponds to the scalar product of the null generators at the intersection whenever the data happens to be embedded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then, the gauge-covariant definition of double null data consists of two null hypersurface data with their boundaries identified via a ∂-isometry and satisfying the compatibility conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' When the double null data happens to be embedded, this notion corresponds to two transverse, null hypersurfaces, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The compatibility conditions are necessary for any double null data to be embeddable in some ambient spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The question of whether these conditions are also sufficient arises naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We answer this in the affirmative by showing that given double null data there always exists a spacetime where it can be embedded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Obviously there are many spacetimes where a given double null data can be embedded and they are in gen- eral not solutions of any field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In that sense, the compatibility conditions are of geometric nature, and they need to be present regardless the spacetime one wants to construct is a solution of the Einstein equations (or any other field equations) or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' If in addition the abstract constraint equations hold, we proved in [13] that the double null data can be embedded in a spacetime solution of the Λ-vacuum Einstein equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Our strategy there was to work in the so-called harmonic gauge (which is still diffeomor- phism covariant) and solve the reduced Einstein equations from the metric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Once the spacetime is constructed we built new embedded data and show that (i) such embed- ded data coincides with the original one and (ii) that the spacetime is a solution of the Einstein equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Thus, there is a clear hierarchy between the compatibility conditions and the constraint equations, in the sense that the former are purely geometric and the later depend on the field equations one wants to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The other central question we want to address in this paper concerns the geometric uniqueness of the characteristic problem at the abstract level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In the standard Cauchy problem, it is known that two initial data sets (Σ, h, K) and (Σ′, h′, K′) such that (Σ, h) and (Σ′, h′) are isometric and the isometry maps K′ into K, give rise to two isometric spacetimes (M, g) and (M′, g′) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' To find a result of this type in the characteristic case we need to give a notion of isometry between two abstract initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' To do that it is essential to take into account the gauge freedom present in the hypersurface data formalism, since two abstract hypersurface data D and D′ related by a gauge transforma- tion are indistinguishable from a geometric point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Hence, the notion of isometric double null data involves a diffeomorphism between the two abstract hypersurfaces (and their corresponding abstract tensors) as well as a a gauge transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' With this definition at hand, we prove that two double null data satisfying the constraint equa- tions are isometric (in the abstract sense) if and only if the spacetimes they define are isometric (in the standard sense).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 3 This paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In Section 2 we recall the basic definitions and results of hypersurface data from [11, 10], such as the notion of hypersurface data, embeddedness and gauge transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We also summarize definitions and results from [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In Section 3 we study non-degenerate codimension-one submanifolds of hypersurface data, showing that the notion of second fundamental forms and torsion one-forms of a submanifold can be defined abstractly from hypersurface data and a new object called normal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In Section 4 we give a fully diffeomorphism and gauge covariant definition of double null data, thus generalizing our previous definition in [13], and we show that the compatibility conditions are necessary and sufficient for a double null data to be embeddable in some spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' To do that we need the explicit expression of the pullback of the constraint tensor into the boundary, which is obtained in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Finally, in Section 5, we study the necessary conditions for two different initial data to define isometric spacetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' This conditions lead to the definition of isometric double null data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The paper finishes with the proof that two isometric double null data define isometric spacetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' This gives a geometric uniqueness notion of the characteristic initial value problem in a fully abstract way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Notation and conventions Let H be a smooth manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We denote by F(H) the set of smooth real functions on H, and by F ⋆(H) the subset of nowhere-vanishing functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The tangent (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' cotangent) bundle of H is denoted by TH (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' T ⋆H), and the set of vector fields (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' covector fields) is X(H) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' X⋆(H)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The interior contraction of a tensor T with a vector X is ιXT := T(X, · · ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Given a diffeomorphism φ : N −→ M and a vector field X ∈ X(M), the pullback of X is given by φ⋆X := (φ−1)⋆X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We employ Greek letters (µ, ν, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=') for spacetime abstract indices, small Latin letters from the beginning of the alphabet (a, b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=') for abstract indices on H and capital Latin letters from the beginning of the alphabet (A, B, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=') for abstract indices on a section of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Given a map ϕ : M −→ N and a submanifold S of M, we denote by ϕ|S : S −→ ϕ(S) the restriction of ϕ to S, both in the domain and the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' As usual, parenthesis (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' brackets) denote symmetrization (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' antisymmetrization) of indices, and ⊗s is the symmetrized tensor product, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=', T1 ⊗s T2 := 1 2 (T1 ⊗ T2 + T2 ⊗ T1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Our convention for the curvature tensor of a connection ∇ is R(X, Y )Z = ∇X∇Y Z − ∇Y ∇XZ − ∇[X,Y ]Z, and its Ricci tensor is the contraction between its contravariant index and its second covariant index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In this paper all manifolds are connected unless otherwise indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 2 Preliminaries In this section we summarize the hypersurface data formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Details can be found in [10, 11] and its precursor [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Our main interest is the characteristic case, so we shall also recall from our previous paper [13] the necessary definitions and results for this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let H be a smooth m-dimensional manifold, γ a symmetric two-covariant tensor field, ℓ a one-form and ℓ(2) a scalar on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' A four-tuple Dmet = {H, γ, ℓ, ℓ(2)} defines metric hypersurface data (of dimension m) provided that the symmetric two- covariant tensor A|p on TpH × R defined by A|p ((W, a), (Z, b)) := γ|p(W, Z) + aℓ|p(Z) + bℓ|p(W) + abℓ(2)|p 4 is non-degenerate at every p ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' A five-tuple D = Dmet∪{Y}, where Y is a symmetric two-covariant tensor field on H, is called hypersurface data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Since the tensor A is non-degenerate one can define its contravariant “inverse” version by A♯ (A ((V, a), ·) , ·) = (V, a) for every (V, a) ∈ X(H) × F(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' From A♯ one defines the two-contravariant, symmetric tensor field P, the vector n and the scalar function n(2) on H by means of A♯ ((α, a), (β, b)) = P(α, β) + an(β) + bn(α) + abn(2), for all α, β ∈ X⋆(H) and a, b ∈ F(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Alternatively, P, n and n(2) can be defined by γ(n, ·) + n(2)ℓ = 0, (1) ℓ(n) + n(2)ℓ(2) = 1, (2) P(ℓ, ·) + ℓ(2)n = 0, (3) P (γ(X, ·), ·) = X − ℓ(X)n ∀X ∈ X(H), (4) γ (P(α, ·), ·) = α − α(n)ℓ ∀α ∈ X⋆(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (5) Despite its name, the abstract manifold H in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='1 is not a hypersurface of any ambient space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The connection with the standard notion of hypersurface is given in the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' A metric hypersurface data Dmet = {H, γ, ℓ, ℓ(2)} is embedded in a semi-Riemannian manifold (M, g) if there exists an embedding Φ : H ֒→ M and a vector field ξ along Φ(H) everywhere transversal to Φ(H), called rigging, such that Φ⋆(g) = γ, Φ⋆ (g(ξ, ·)) = ℓ, Φ⋆ (g(ξ, ξ)) = ℓ(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (6) The hypersurface data D = {H, γ, ℓ, ℓ(2), Y} is embedded provided that, in addition, 1 2Φ⋆ (Lξg) = Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (7) As usual, we define the radical of γ by Rad(γ) := {X ∈ X(H) such that γ(X, ·) = 0} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' It can be shown that the vector space Rad(γ)p at each p ∈ H is at most one-dimensional [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then, from equation (1), the condition n(2) = 0 is equivalent to Rad(γ) = span {n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' When the data is embedded, the first relation in (6) together with Rad(γ) ̸= {0} imply that Φ(H) is a null hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Motivated from this geometric picture, hypersurface data satisfying n(2) = 0 everywhere on H will be called null hypersurface data (NHD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' A smooth submanifold S ⊂ H is called a section of H provided that every integral curve of n intersects transversely S exactly once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let D be embedded hypersurface data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Given a rigging ξ, any other vector of the form ξ′ = z(ξ + ζ), with z ∈ F ⋆(H) and ζ ∈ X(H), is again a rigging vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' At the abstract level, this freedom is encoded in the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 5 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let D = {H, γ, ℓ, ℓ(2), Y} be hypersurface data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let z ∈ F ⋆(H) and ζ ∈ X(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The gauge transformed hypersurface data with gauge parameters (z, ζ) are G(z,ζ) (γ) := γ, (8) G(z,ζ) (ℓ) := z (ℓ + γ(ζ, ·)) , (9) G(z,ζ) � ℓ(2)� := z2 � ℓ(2) + 2ℓ(ζ) + γ(ζ, ζ) � , (10) G(z,ζ) (Y) := zY + ℓ ⊗s dz + 1 2Lzζγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (11) The gauge transformation laws (8)-(10) induce the following transformations on the contravariant data [10] G(z,ζ) (P) = P + n(2)ζ ⊗ ζ − 2ζ ⊗s n, (12) G(z,ζ) (n) = z−1(n − n(2)ζ), (13) G(z,ζ) � n(2)� = z−2n(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (14) We will often employ a prime to denote the gauge transformed objects when the gauge group element is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The set of gauge transformations defines a group with composition law and inverse given by [10] G(z1,ζ1) ◦ G(z2,ζ2) = G(z1z2,ζ2+z−1 2 ζ1) (15) G−1 (z,ζ) = G(z−1,−zζ), (16) and neutral element G(1,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Given hypersurface data we define the following two-covariant tensor fields F := 1 2dℓ (17) and K := n(2)Y + 1 2Lnγ + ℓ ⊗s dn(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (18) When the data is embedded the tensor K corresponds [11] to the second fundamental form of Φ(H) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' the unique normal vector ν satisfying g(ν, ξ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' As expected from this geometric interpretation, the transformation law of K is G(z,ζ) (K) = z−1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (19) Given hypersurface data, a unique torsion-free connection ∇ exists with the following defining properties � ∇Xγ � (Z, W) = −ℓ(Z)K(X, W) − ℓ(W)K(X, Z), (20) � ∇Xℓ � (Z) = Y(X, Z) + F(X, Z) − ℓ(2)K(X, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (21) Under a gauge transformation with gauge parameters (z, ζ), the connection ∇ transforms as [13] G(z,ζ) � ∇ � = ∇ + ζ ⊗ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (22) The combination Y + F will appear frequently below, so we give it a name, Π := Y + F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (23) 6 The tensor Π has the following interesting property [13, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (36)] Π(n, X) − Π(X, n) = (Lnℓ) (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (24) The transformation law of Π(·, n) is [13, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (38)] Π′(·, n′) = Π(·, n) − K(·, ζ) + d log |z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (25) A consequence of (20)-(21) together with (1)-(4) is [10] ∇Xn = P � K(X, ·) − n(2)Π(X, ·), · � − � Π(X, n) + n(2)X � ℓ(2)�� n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (26) For null hypersurface data (n(2) = 0) the previous equation takes the simpler form ∇Xn = P (K(X, ·), ·) − Π(X, n)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (27) The connection ∇ has the following geometric interpretation in the embedded case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Given Dmet with embedding Φ and rigging ξ in an ambient manifold (M, g) with Levi– Civita connection ∇ and X, Z ∈ X(H), the relation between ∇ and ∇ is [10, 12] ∇XZ = ∇XZ − K(X, Z)ξ, (28) where we have identified vector fields in X(H) and their image under Φ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' This slight abuse of notation will be used repeatedly from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The rest of this section is devoted to summarizing the results of characteristic hypersur- face data developed in [13] needed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let D = {H, γ, ℓ, ℓ(2), Y} be hypersurface data of dimension m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We say that the set D is “characteristic hypersurface data” (CHD) provided that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Rad(γ) ̸= {0} and γ is semi-positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' There exists u ∈ F(H) satisfying λ := n(u) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Such functions are called “folia- tion functions” (FF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The leaves Su := {p ∈ H : u(p) = u} are all diffeomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Henceforth, we will assume H to have boundary of the form ∂H = {u = u0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Given CHD, the existence of such foliation function allows us to define a tangent space decomposition at every p ∈ H of the form TpH = TpSu(p) ⊕ span {n|p} , (29) where TpSu(p) = {X ∈ TpH such that X(u) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' This decomposition induces an- other on the cotangent space given by T ⋆ p H = T ⋆ p Su(p) ⊕ span {du|p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then the tangent (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' cotangent) bundle decomposes as the direct sum TH = TS ⊕ span {n} (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' T ⋆H = T ⋆S ⊕ span {du}), where TS = � TqSu, and therefore every X ∈ X(H) can be written uniquely as X = X∥ + σXn with X∥ ∈ X(S) and σX ∈ F(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' A vector field X is said to be tangent to the foliation provided σX = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let D be CHD, u a foliation function and i : Su ֒→ H the inclusion of each leaf onto H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' For each Su we define h := i⋆γ, the one-form ℓ∥ := i⋆ℓ ∈ X⋆(Su), (30) 7 and the vector ℓ♯ := h♯(ℓ∥, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (31) It is immediate to show that h is a positive definite metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The transformation laws of ℓ∥ and ℓ♯ follow directly from that of ℓ (see (9)) ℓ′ ∥ = z � ℓ∥ + h(ζ∥, ·) � , (32) ℓ′♯ = z(ℓ♯ + ζ∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (33) In [13] the following set of so-called “foliation tensors” were introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' By construction they are intrinsic to the leaves of the foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let D be CHD and u a foliation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We define the “foliation tensors” χ, Υ, τ and η on each leaf Su by χ := i⋆K, Υ := i⋆Y, τ := i⋆ (Π(·, n) + ιℓ♯K) η := −τ − d(log |λ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In the embedded case, χ is the second fundamental form of Su along the normal vector ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The tensors Υ and η do not admit such a neat geometric interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' However, in an appropriate gauge (namely such that the rigging is null and normal to Su), Υ coincides with the null second fundamental form along ξ and η is the torsion one-form of Su w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='t the null basis {ν, ξ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Finally we recall that Y(n, n) measures the deviation of ν from being geodesic (this follows exclusively from equations (21), (28) and therefore it does not depend on the gauge or on the existence of a foliation on H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We refer the reader to [13, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 5] for the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' As shown in [13], given embedded null hypersurface data in a spacetime (M, g), all tangential components of the ambient Ricci tensor on the hypersurface can be written in terms of the abstract data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' This allowed us to define the abstract constraint tensor Rab by Rab := � γafR f cbd + 2∇[d � Kb]cℓa � + 2ℓ(2)Kc[bKd]a � P cd (34) − � ℓdR d bac + 2ℓ(2)∇[cKa]b + Kb[a∇c]ℓ(2) + ℓdR d abc + 2ℓ(2)∇[cKb]a + Ka[b∇c]ℓ(2)� nc, where R a bcd is the curvature tensor of ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' As shown in [13], this tensor satisfies R = Φ⋆ Ric, (35) where Ric is the Ricci tensor of (M, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' This property suggests that R is gauge invariant and, indeed, in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='6 in [13] we have established G(z,ζ) (R) = R ∀(z, ζ) ∈ F ⋆(H) × X(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (36) By the geometric interpretation of this tensor, the condition R = 0 can be thought as the vacuum constraint equations on a null hypersurface H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' These equations are fully diffeomorphism and gauge covariant and do not assume the presence of any foliation, thus generalizing the so-called null structure equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Recall that the hypersurface data formalism allows us to write these constraint equations even though there is no metric on H, thanks to the existence of the abstract connection ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 8 In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 6 of [13] we introduced a fully covariant gauge which translates the harmonic condition on the ambient coordinates into an abstract condition on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Given a set of independent functions {xa} on H and a (local) basis {ec} of X(H), we define Bca := ec(xa) and Bca as its inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Since, for each value of a, Bca is a vector so it is V c := Bca□Pxa, where □P f := P ab∇a∇bf for every scalar f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' This vector allows us to define the so-called harmonic gauge, which one can show it exists and is unique up to the choice of the gauge parameters at a given “initial” section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We refer the reader to Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='2 of [13] for the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let D be null hypersurface data admitting a section S ⊂ H, {xa} a set of m functionally independent functions, V c := Bca□P xa and trP K := P abKab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then there exists a class of gauges satisfying the following conditions on H, trP K − 2Y(n, n) = 0, (37) 2P � ∇nℓ, · � + n � ℓ(2)� n = V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (38) The set of transformations keeping the previous conditions invariant is parametrized by (z0, ζ0) ∈ F ⋆(S) × X(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Any gauge satisfying (37) and (38) will be called “harmonic gauge” (HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The name “harmonic” is motivated by the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Given embedded CHD in a spacetime (M, g) with rigging ξ and embedding Φ, we can extend the functions xa off Φ(H) by ξ(xa) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Moreover, we can introduce another function u on M such that u|H = 0 and ξ(u) = 1 on Φ(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' As shown in [13], the data is written in the harmonic gauge if and only if the functions {u, xa} satisfy □gxa = □gu = 0 on Φ(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' One can exploit the remaining gauge freedom in the class of gauges of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='6 to fix the values of ℓ(2) and ℓ∥ to zero in any section S (see [13, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let D be null hypersurface data admitting a section S ⊂ H, {xa} a set of m functionally independent functions and V c := Bca□Pxa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then there exists a class of gauges satisfying (37)-(38) on H as well as ℓ(2) = 0, ℓ∥ = 0 on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The set of transformations keeping the previous conditions invariant is parametrized by z0 ∈ F ⋆(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let {xa} = {u, xA} be functions on H satisfying (i) n(u) ̸= 0, (ii) n(xA) = 0 and (iii) {xA} is a coordinate system on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' When the data is written in the gauge of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='7 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='t {xa}, conditions (38) can be rewritten as equations involving the Y tensor (see [13, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let D be null hypersurface data admitting a section S written in the gauge of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='7 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='t some functions xa = {u, xA} satisfying conditions (i), (ii) and (iii) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then the following equations hold at S trh Υ := hABΥAB = n � ℓ(2)� , (39) 2 (Lnℓ + Π(·, n)) (gradhxA) = □hxA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (40) 3 Non-degenerate submanifolds In this section we study embedded non-degenerate submanifolds in the context of hy- persurface data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' For the purposes of this paper we restrict ourselves to the null case but 9 without fixing a priori the topology of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let D = {H, γ, ℓ, ℓ(2), Y} be null hypersurface data of dimension m and Σ an (m − 1)-dimensional manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We say that i : Σ ֒→ H is a non-degenerate submanifold (of codimension one) of D provided that h := i⋆γ is non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The idea is to identify, in terms of hypersurface data, a suitable notion of the torsion and second fundamental form of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We start by assuming that D is em- bedded null hypersurface data with rigging ξ and embedding Φ in a spacetime (M, g) with Levi-Civita connection ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Given a vector field t along Φ (i (Σ)) everywhere normal to Σ, there exist a function C ∈ F(Σ) and a vector field T ∈ X(H) such that t = Cξ + Φ⋆T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (41) By (6), the condition that t is normal to Σ is equivalent to Cℓ(X) + γ(T, X) = 0 ∀X ∈ X(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Given X, Y ∈ X(Σ), the (ambient) second fundamental form IIt of Σ along the normal vector t can be defined as IIt(X, Y ) = 1 2 (Ltg) (X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Substituting the expression (41) and employing the simple formula LfV g = fLV g + 2df ⊗s g(V, ·) (42) valid for any function f and vector field V , IIt(X, Y ) = 1 2 (LΦ⋆Tg) (X, Y ) + 1 2C (Lξg) (X, Y ) + (dC ⊗s g(ξ, ·))(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Inserting the expressions of Y and ℓ in the context of embedded hypersurface data (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='2) and using Φ⋆ (LΦ⋆Tg) = LT (Φ⋆g) = LTγ, one finally gets IIt(X, Y ) = 1 2 (LTγ) (X, Y ) + CY(X, Y ) + 1 2 (X(C)ℓ(Y ) + Y (C)ℓ(X)) , (43) which is an expression only depending on (abstract) hypersurface data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In addition to the second fundamental form, the geometry of submanifolds of (ambient) codimension two also consist of the torsion one-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Given a basis of normal vectors {ti = Ciξ + Φ⋆Ti}, i = 1, 2, the (ambient) torsion of Σ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' this basis is the set of one-forms Tij given by Tij(X) = g(ti, ∇Xtj) ∀X ∈ X(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (44) As in the case of the second fundamental form, this object can be also rewritten in terms of hypersurface data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Indeed, from (41) and (28) ∇Xtj = ∇X (Cjξ + Φ⋆Tj) = (X(Cj) − K(X, Tj)) ξ + Cj∇Xξ + ∇XTj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (45) Then, using (6) g(ξ, ∇Xtj) = (X(Cj) − K(X, Tj)) ℓ(2) + 1 2CjX(ℓ(2)) + ℓ(∇XTj), (46) and g(Φ⋆Ti, ∇Xtj) = (X(Cj) − K(X, Tj)) ℓ(Ti) + Cjg(Φ⋆Ti, ∇Xξ) + γ(Ti, ∇XTj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 10 From equations (28), (21) and the definition of ℓ in the context of embedded data (Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='2), g(Φ⋆Ti, ∇Xξ) = X (g(Φ⋆Ti, ξ)) − g (∇X(Φ⋆Ti), ξ) = X (g(Φ⋆Ti, ξ)) − g � ∇XTi, ξ � + K(X, Ti)g(ξ, ξ) = X (ℓ(Ti)) − ℓ � ∇XTi � + ℓ(2)K(X, Ti) = � ∇Xℓ � (Ti) + ℓ(2)K(X, Ti) = Π(X, Ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then, g(Φ⋆Ti, ∇Xtj) = (X(Cj) − K(X, Tj)) ℓ(Ti) + CjΠ(X, Ti) + γ(Ti, ∇XTj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (47) Introducing (45) into (44) and employing (46) and (47), Tij(X) = g(Ciξ + Φ⋆Ti, ∇Xtj) = � ℓ(Ti) + Ciℓ(2)� (X(Cj) − K(X, Tj)) + Ciℓ � ∇XTj � + γ(Ti, ∇XTj) + 1 2CiCjX � ℓ(2)� + CjΠ(X, Ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (48) The computation above suggests introducing a fully abstract notion of normal vector to the submanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' This notion is called “normal pair” (NP) and it is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let D = {H, γ, ℓ, ℓ(2), Y} be hypersurface data and i : Σ ֒→ H a non- degenerate submanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' A normal pair t := {T, C} consists of a vector field T ∈ X(H) along i(Σ) and a function C ∈ F(Σ) satisfying Cℓ(X) + γ(T, X) = 0 ∀X ∈ X(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (49) Since Σ is a codimension one submanifold of H, Σ admits exactly two linearly indepen- dent NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Indeed, equation (49) can be written in terms of the tensor A (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='1) as A ((T, C), (X, 0)) = 0 for every X ∈ X(Σ) along i(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Since A|p is non-degenerate at every p ∈ Σ, the (m+1)- dimensional vector space TpH × R can be decomposed as TpH × R = TpΣ ⊕ (TpΣ)⊥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Since TpΣ has dimension (m − 1) it follows that (TpΣ)⊥ is two-dimensional and hence Σ admits exactly two linearly independent NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Motivated by (43), given a normal pair t = {T, C}, the second fundamental form of Σ along t is the tensor field on Σ defined by Kt(X, Y ) := 1 2 (LTγ) (X, Y ) + CY(X, Y ) + 1 2 (X(C)ℓ(Y ) + Y (C)ℓ(X)) , (50) for X, Y ∈ X(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In the embedded case, given a normal pair t = {C, T} we can associate to it the (ambient) vector field t[t] defined by t[t] := Cξ + Φ⋆T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' By construction, t[t] is normal to Φ(i(Σ)), since g(t[t], X) = Cℓ(X) + γ(T, X) = 0 for every X ∈ X(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Comparing the expression of the (ambient) second fundamental form IIt[t] of Φ (i(Σ)) in (43) with the abstract Kt in definition (50), the following result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 11 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let D = {H, γ, ℓ, ℓ(2), Y} be embedded hypersurface data on a space- time (M, g) with embedding Φ and rigging ξ, and let Σ be a non-degenerate submanifold of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let t = {T, C} be a NP of Σ and let t = Cξ + Φ⋆T be its associated (ambient) vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then, Kt(X, Y ) = IIt[t](X, Y ) for every X, Y ∈ X(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' As expected from the spacetime interpretation, Kt has the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let t = {T, C} be a normal pair, Kt the second fundamental form as in (50) and f a scalar function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then, Kft = fKt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Directly from the definition (50) and the formula (42), Kft(X, Y ) = 1 2 (LfTγ) (X, Y ) + fCY(X, Y ) + (d (fC) ⊗s ℓ) (X, Y ) = fKt(X, Y ) + (df ⊗s γ(T, ·)) (X, Y ) + (Cdf ⊗s ℓ) (X, Y ) = fKt(X, Y ) + (df ⊗s (γ(T, ·) + Cℓ)) (X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Since t is a normal pair, the term in parenthesis in the last line vanishes when contracted with tangent vectors, and thus result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let {ti} be a basis of normal pairs and ˆti = Ωj itj a change of basis, with Ωj i ∈ F(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then, K ˆti = Ωj iKti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' By the previous Lemma it suffices to show that Kt1+t2 = Kt1 + Kt2 for every pair of NPs t1 = (T1, C1) and t2 = (T2, C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Directly from the definition of K in (50), Kt1+t2(X, Y ) = 1 2 (LT1+T2γ) (X, Y ) + (C1 + C2)Y(X, Y ) + 1 2 (X(C1 + C2)ℓ(Y ) + Y (C1 + C2)ℓ(X)) = Kt1(X, Y ) + Kt2(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Finally, Kt1+t2 = Kt1 + Kt2 together with Kft = fKt proves K ˆti = Ωj iKti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The spacetime picture also motivates the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let D = {H, γ, ℓ, ℓ(2), Y} be hypersurface data and i : Σ ֒→ H a non- degenerate submanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let (z, ζ) be a gauge group element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The gauge transformation of a normal pair t = {T, C} of Σ is defined as G(z,ζ)(t) := {T ′ = T − Cζ, C′ = z−1C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Note that t′ = G(z,ζ)(t) is still a normal pair, since C′ℓ′(X) + γ(T ′, X) = Cℓ(X) + Cγ(ζ, X) + γ(T, X) − Cγ(ζ, X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 12 Moreover, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='5 is a realization of the gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Indeed, let (z1, ζ1), (z2, ζ2) be gauge parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' On the one hand, G(z1,ζ1)G(z2,ζ2){T, C} = G(z1,ζ1) � T − Cζ2, z−1 2 C � = � T − (z−1 2 ζ1 + ζ2)C, z−1 1 z−1 2 C � , and on the other, using (15), G(z1,ζ1)G(z2,ζ2){T, C} = G(z1z2,ζ2+z−1 2 ζ1) {T, C} = � T − (z−1 2 ζ1 + ζ2)C, z−1 1 z−1 2 C � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' As expected from the spacetime picture, the second fundamental form along a normal pair is gauge invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let D be hypersurface data, Σ a non-degenerate submanifold and t a normal pair of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then Kt is gauge invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let t = {T, C} and t′ = {T ′ = T − Cζ, C′ = z−1C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Using the transformation laws (9) and (11) (we drop the argument (X, Y ) in order not to overload the notation) Kt′ = 1 2LT ′γ + C′Y′ + dC′ ⊗s ℓ′ = 1 2LTγ − 1 2CLζγ − dC ⊗s γ(ζ, ·) + CY + z−1Cℓ ⊗s dz + 1 2CLζγ + z−1Cdz ⊗s γ(ζ, ·) + dC ⊗s (ℓ + γ(ζ, ·)) + zCdz−1 ⊗s (ℓ + γ(ζ, ·)) = 1 2LTγ + CY + dC ⊗s ℓ = Kt, where in the second line we used the formula (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let i : Σ ֒→ H be a non-degenerate submanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Following the notation in (30)-(31) in the context of CHD, we define the one-form ℓ∥ ∈ X⋆(Σ) and the vector ℓ♯ ∈ X(Σ) by means of ℓ∥ := i⋆ℓ, ℓ♯ := h♯(ℓ∥, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (51) In the next proposition we identify a particular relevant normal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let D = {H, γ, ℓ, ℓ(2), Y } be null hypersurface data and i : Σ ֒→ H a non-degenerate submanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Define the vector θ along i(Σ) by θ := −1 2 � ℓ(2) − h � ℓ♯, ℓ♯�� n − i⋆ℓ♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (52) Then, tθ := (θ, 1) is a normal pair of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Furthermore, tθ is A-null, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=', A(tθ, tθ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' For any X ∈ X(Σ), A ((i⋆X, 0), (θ, 1)) = ℓ(i⋆X) + γ(i⋆X, θ) = ℓ∥(X) − h(X, ℓ♯) = 0, where we used γ(n, ·) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Hence, tθ is a NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Moreover, A ((θ, 1), (θ, 1)) = γ(θ, θ) + 2ℓ(θ) + ℓ(2) = h(ℓ♯, ℓ♯) − ℓ(2) + h � ℓ♯, ℓ♯� − 2ℓ∥(ℓ♯) + ℓ(2) = 0, where now we used γ(n, ·) = 0 and ℓ(n) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 13 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Observe that the normal pair tn := (n, 0) is linearly independent of tθ and also satisfies A(tn, tn) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In fact, since the space of normal pairs is two-dimensional and A(tn, tθ) = γ(n, θ) + ℓ(n) = 1, it follows that any normal pair which is also A-null lies either in span{tn} or span{tθ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' From equation (48) the following abstract definition of the torsion one-forms is motivated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let D = � H, γ, ℓ, ℓ(2), Y � be null hypersurface data and {ti = (Ti, Ci)} with i = 1, 2, a basis of normal pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The torsion one-forms of Σ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='t this basis are the tensors on Σ defined by ℸij(X) := � ℓ(Ti) + Ciℓ(2)� (X(Cj) − K(X, Tj)) + Ciℓ � ∇XTj � + γ(Ti, ∇XTj) + 1 2CiCjX � ℓ(2)� + CjΠ(X, Ti) for every X ∈ X(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We will also employ the notation ℸ(ti, tj) when ℸij causes confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let {ti, tj} be a basis of normal pairs and let ti[ti], tj[tj] be their associated (ambient) vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Comparing the expression of the (ambient) torsion one-forms Tij of Φ (i(Σ)) in (44) with the abstract ℸij in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='9, the following result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let D = {H, γ, ℓ, ℓ(2), Y} be embedded hypersurface data on a space- time (M, g) with embedding Φ and rigging ξ, and let Σ be a non-degenerate submanifold of H with a basis of normal pairs {ti}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then, ℸij(X) = Tij(X) for every X ∈ X(Σ), where Tij are the ambient torsion one-forms w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='t the basis {ti[ti]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Since Σ is a codimension-one submanifold of H one has in principle four different torsion one-forms ℸij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' However, not all them are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let D = � H, γ, ℓ, ℓ(2), Y � be null hypersurface data, {(Ti, Ci)} a basis of normal pairs and Mij := A ((Ti, Ci), (Tj, Cj)) = γ(Ti, Tj) + Ciℓ(Tj) + Cjℓ(Ti) + CiCjℓ(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (53) Then, ℸij + ℸji = dMij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Directly from the definition of ℸ, ℸij(X) + ℸji(X) = X � CiCjℓ(2)� + 2 � X(C(i) − K(X, T(i) � ℓ(Tj)) − 2ℓ(2)C(iK(X, Tj)) + 2C(iℓ � ∇XTj) � + 2γ � T(i, ∇XTj) � + 2C(iΠ(X, Tj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (54) Using equation (20), 2γ � T(i, ∇XTj) � = X (γ(Ti, Tj)) − � ∇Xγ � (Ti, Tj) = X (γ(Ti, Tj)) + 2K(X, T(i)ℓ(Tj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (55) From (21), 2C(iℓ � ∇XTj) � = 2X � C(iℓ(Tj)) � − 2X(C(i)ℓ(Tj)) − 2C(i � ∇Xℓ � (Tj)) = 2X � C(iℓ(Tj)) � − 2X(C(i)ℓ(Tj)) − 2C(iΠ(X, Tj)) + 2ℓ(2)C(iK(X, Tj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (56) Inserting (55) and (56) into (54) yields the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 14 In fact, since Σ is a codimension one submanifold of H, there is only one independent torsion one-form on Σ, that can be taken to be ℸ12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In the embedded case, given two normal pairs ti and tj, the function Mij = A (ti, tj) as defined in (53) corresponds to the scalar product g(ti, tj), where ti and tj are the ambient vector fields associated to ti and tj, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' As expected from this geometric interpretation, the functions Mij are gauge invariant, as we show next at the abstract level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let {ti} be a basis of normal pairs of a non-degenerate submanifold Σ and Mij := A (ti, tj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then, the functions Mij are gauge invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let (z, ζ) be gauge parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Directly from (53) and Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='5, M′ ij = A′ � (Ti − Ciζ, z−1Ci), (Tj − Cjζ, z−1Cj) � = γ (Ti − Ciζ, Tj − Cjζ) + z−1Ciℓ′ (Tj − Cjζ) + z−1Cjℓ′ (Ti − Ciζ) + z−2CiCjℓ(2)′ = γ (Ti, Tj) + Ci � z−1ℓ′(Tj) − γ(ζ, Tj) � + Cj � z−1ℓ′(Ti) − γ(ζ, Ti) � + CiCj � z−2ℓ(2)′ + γ(ζ, ζ) − 2z−1ℓ′(ζ) � = γ(Ti, Tj) + Ciℓ(Tj) + Cjℓ(Ti) + CiCjℓ(2) = Mij, where in the fourth line we used (8)-(10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' As expected from the geometric interpretation of the ambient torsion one-forms as the connection coefficients of the normal bundle connection, the transformation of the ab- stract torsion one-forms under a change of basis of normal pairs is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let {ti} be a basis of normal pairs of Σ and ˆtj = Ωi jti with Ωi j ∈ F(Σ) a change of basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then, �ℸij = Ωk i Ωl jℸkl + Ωk i Mkl dΩl j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (57) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Writing ti = (Ti, Ci) and ˆti = ( �Ti, �Ci), the change of basis gives �Ci = Ωk i Ck, �Ti = Ωk i Tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Inserting them in the expression of ℸij and noting that all the terms are multilinear except for X( �Cj) and ∇X �Tj, which become X( �Cj) = Ωl jX(Cl) + ClX(Ωl j), ∇X �Tj = Ωl j∇XTl + X(Ωl j)Tl, one immediately gets �ℸij(X) = Ωk i Ωl jℸkl + Ωk i � CkClℓ(2) + Ckℓ(Tl) + Clℓ(Tk) + γ(Tk, Tl) � X � Ωl j � , which is (57) after recalling the definition of Mkl in (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let t1 and t2 two linearly independent normal pairs and f a scalar function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Under a change of basis {t1, t2} �−→ {t′ 1 = f −1t1, t′ 2 = ft2}, the torsion one- form ℸ(t1, t2) transforms as �ℸ12 = ℸ12 + M12d log |f|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In the next proposition we show that the abstract torsion one-forms are gauge invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 15 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let D be hypersurface data, Σ a non-degenerate submanifold and {ti = (Ti, Ci)} a basis of normal pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then, the torsion one-forms ℸij are gauge invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' First we show that if ℸij are gauge invariant in a particular basis of normal pairs, they are also invariant in any basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let {ti} (with i = 1, 2) be a basis of normal pairs and let ˆtj = Ωi jti with Ωi j ∈ F(Σ) be a change of basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' From the transformation law in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='5, �Cj = Ωi jCi, �Tj = Ωi jTi =⇒ z−1 �Cj = Ωi jz−1Ci, �Tj − �Cjζ = Ωi j (Ti − Ciζ) , which means that the gauge transformed basis {t′ i} and {ˆt′ i} are related by ˆt′ j = Ωi jt′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Thus, the functions Ωi j are gauge invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Moreover, from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='12 the functions Mij are gauge invariant too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then, from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='13, if ℸij is gauge invariant, so it is �ℸij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Hence, it suffices to show the statement in the basis {tn, tθ}, where the only non-zero torsion one-forms are given by ℸnθ(X) = −ℸθn(X) = −K(X, θ) + Π(X, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' From Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='5, given gauge parameters (z, ζ) the transformed basis of normal pairs is G (tn) = (n, 0) and G (tθ) = (θ − ζ, z−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Hence, applying Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='9 to the new gauge, G (ℸnθ) (X) = z � X(z−1) − K′(X, θ − ζ) � + z−1Π′(X, n) = −X (log |z|) − K(X, θ) + K(X, ζ) + Π′(X, n′) = −X (log |z|) − K(X, θ) + K(X, ζ) + Π(X, n) + X (log |z|) − K(X, ζ) = ℸnθ(X), where in the third line we used the transformation law of Π(·, n) in (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In the context of CHD the leaves {iu : Su ֒→ H} are non-degenerate submanifolds of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then, it is natural to connect the geometry of the foliation in a CHD with the tools developed in this Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let D be CHD, u a foliation function and Su any leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' By Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='8, tn = (n, 0) and tθ = (θ, 1) constitute a basis of A-null, normal pairs of Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Using (50) and Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='5, we have Ktn = χ and Ktθ = Υ + Θ, where Θ := 1 2i⋆ u (Lθγ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (58) From the definition of the torsion one-form (Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='9) one also has ℸnθ(X) = −K(X, θ) + Π(X, n) = K(X, ℓ♯) + Π(X, n) = τ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Thus, the tensors χ and Υ+Θ are the second fundamental forms w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='t the normal pairs tn = (n, 0) and tθ = (θ, 1), respectively, whereas the tensor τ is the torsion one-form w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='t the basis {tn, tθ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 4 Gauge-covariant compatibility conditions In Section 7 of [13] we studied the necessary conditions that two CHD must fulfil in order to be simultaneously embedded in the same spacetime with common spacelike boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' However, these compatibility conditions were obtained in a particular gauge 16 where strong restricting conditions were imposed at the respective boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In this Section we generalize the compatibility conditions and write them employing the tools developed in the previous section, which allows us to define the notion of double null data in a fully gauge-covariant way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We start introducing the key notion that will allow us to do that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let D and D be two null hypersurface data with boundaries S := ∂H and S := ∂H and φ : S −→ S a diffeomorphism between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' An invertible linear map Ψp : TpH ⊕ R −→ Tφ(p)H ⊕ R is called a ∂-isometry between CHD at p ∈ S provided that1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Ψp|TpS ((X, 0)) = φ⋆|p(X) for all X ∈ X(S), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Ψ⋆ pAφ(p) = Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' If conditions (1) and (2) hold at every p ∈ S we say that D and D are ∂-isometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Since S and S are non-degenerate codimension one submanifolds of H and H, we can talk about normal pairs on S and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Given a NP t = {T, C} of S, the image of t under Ψ is still a normal pair on S, since φ is a diffeomorphism and 0 = A (t, (X, 0)) = (Ψ⋆A) (t, (X, 0)) = A (Ψ(t), (φ⋆X, 0)) ∀X ∈ X(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Moreover, if a normal pair t of S is A-null, the normal pair Ψ(t) of S is A-null, since A (Ψ(t), Ψ(t)) = A (t, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In the following proposition we show that given a diffeomorphism φ : S −→ S and a non-vanishing function µ ∈ F ⋆(S), a natural ∂-isometry can be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let D and D be two NHD and φ : S −→ S a diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' For each µ ∈ F ⋆(S), there exists a unique ∂-isometry Ψ : TH ⊕ F(S) −→ TH ⊕ F(S) determined by the condition A ((n, 0), Ψ(n, 0)) = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The existence and uniqueness proof will be constructive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=', we will impose conditions (1) and (2) in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='1 and show that there is a unique candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We will then check that this candidate is indeed a ∂-isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let X ∈ X(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We define Ψ ((X, 0)) := (φ⋆X, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In order to define the image of (n, 0) and (θ, 1) under Ψ first observe that since (n, 0) and (θ, 1) are A-null, Ψ ((n, 0)) and Ψ ((θ, 1)) are A-null too, and since A ((n, 0), (θ, 1)) = 1 (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='8) we necessarily have Ψ ((n, 0)) ∈ span {(θ, 1)} and Ψ ((θ, 1)) ∈ span {(n, 0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Imposing the condition A ((n, 0), Ψ(n, 0)) = µ the only option is Ψ ((n, 0)) = µ(θ, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (59) To complete the determination of Ψ we only need to find Ψ ((θ, 1)), which we already know is of the form Ψ ((θ, 1)) = α(n, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The proportionality function α is obtained from A ((n, 0), (θ, 1)) = 1 and its underlined version after imposing item (2) of Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='1 as follows 1 = A ((θ, 1), (n, 0)) = A (Ψ ((θ, 1)) , Ψ((n, 0))) = αµA ((n, 0), (θ, 1)) = αµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 1Given V, W ∈ TpH ⊕ R, we define � Ψ⋆Aφ(p) � (V, W) := Aφ(p) (Ψ(V ), Ψ(W)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 17 Since µ ̸= 0 by hypothesis we conclude Ψ ((θ, 1)) = µ−1(n, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (60) In matrix notation, the expression of Ψ in the decomposition TS ⊕ span {(n, 0), (θ, 1)} (and the corresponding one in the image) is2 Ψ = \uf8eb \uf8ed (φ⋆) 0 0 0 0 µ−1 0 µ 0 \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (61) It is now straightforward to check that this candidate to be a ∂-isometry is indeed a ∂-isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Indeed, conditions (1) and (2) of Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='1 hold by construction and by expression (61) and the fact that φ is a diffeomorphism, we conclude that Ψ is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In the proof of Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='2 we found the expression of Ψ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='t decompositions TpS ⊕ span {(n, 0), (θ, 1)} and Tφ(p)S ⊕ span {(n, 0), (θ, 1)}, namely Ψ = \uf8eb \uf8ed (φ⋆) 0 0 0 0 µ−1 0 µ 0 \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (62) In some situations it may be interesting to have an expression for Ψ in a more general basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let D and D be NHD, φ : S −→ S a diffeomorphism and Ψ the unique ∂-isometry satisfying A ((n, 0), Ψ(n, 0)) = µ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let v, v be transverse vectors to S and S, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then the linear map Ψ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='t decompositions TS ⊕ span {(v, 0), (0, 1)} and TS ⊕ span {(v, 0), (0, 1)} is given by Ψ = \uf8eb \uf8ed (φ⋆) φ⋆v∥ − σµ � ασ−1v∥ + ℓ♯� φ⋆ℓ♯ + µαℓ♯ + σ−1(µαα − µ−1)v∥ 0 µσασ−1 σ−1(µ−1 − µαα) 0 µσ −µα \uf8f6 \uf8f8 , (63) where σ ∈ F ⋆(S), σ ∈ F ⋆(S), v∥ ∈ X(S) and v∥ ∈ X(S) are univocally defined by the decompositions v = σn + v∥ and v = σn + v∥, and where we introduce the functions α = −1 2 � ℓ(2) − h(ℓ♯, ℓ♯) � and α = −1 2 � ℓ(2) − h(ℓ♯, ℓ♯) � to simplify the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We only need to compute the second and third columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Since Ψ ((v, 0)) = σΨ ((n, 0)) + Ψ � (v∥, 0) � , employing (59) together with (52), namely θ = αn − ℓ♯ (we omit the i⋆ in order not to overload the notation), Ψ ((v, 0)) = σµ(θ, 1) + (φ⋆v∥, 0) = � σµασ−1(v − v∥) − σµℓ♯ + φ⋆v∥, σµ � , and hence the second column of (63) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The third column requires computing Ψ ((0, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Decomposing (0, 1) = (θ, 1) − (αn, 0) + (ℓ♯, 0) and using (62) yields Ψ(0, 1) = � µ−1n − µαθ + φ⋆ℓ♯, −µα � = � µ−1σ−1(v − v∥) − µαασ−1(v − v∥) + µαℓ♯ + φ⋆ℓ♯, −µα � , so the third column of (63) is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 2When a map is written in matrix notation we follow the convention that the entries of the i-th column are the coefficients of the image of the i-th vector in the basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 18 Next we study how the map Ψ changes under gauge transformations on D and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let D and D be null hypersurface data and Ψ be a ∂-isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Under gauge transformations on D and D with parameters (z, ζ) and (z, ζ), respectively, Ψ transforms as Ψ′ = G−1 ◦ Ψ ◦ G, (64) where G is the invertible linear map G : TH ⊕ F(S) −→ TH ⊕ F(S) (V, a) �−→ G ((V, a)) := (V + azζ, az) (65) and G : TH ⊕ F(S) −→ TH ⊕ F(S) is defined identically but with all the quantities carrying an underline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' As a consequence, the transformation law of the function µ := A ((n, 0), Ψ(n, 0)) is µ′ = z−1z−1µ, (66) where in order not to overload the notation we denote with the same symbol a function f ∈ F(∂H) and f ◦ φ−1 ∈ F(∂H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' This slight abuse of notation will be used repeatedly from now on when no confusion arises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let D be hypersurface data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' From Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='3 one can write the transformation law for the tensor A in a compact way as A′ ((V, a), (W, b)) = A (G(V, a), G(W, b)) (67) for all (V, a), (W, b) ∈ TH⊕F(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' To show this we use matrix notation in which vectors are represented as columns and covectors as rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The map (65) gets rewritten in matrix form as � V + azζ az � = � 1 zζ 0 z � � V a � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Define, therefore (G) = � 1 zζ 0 z � , (A) = �γ ℓT ℓ ℓ(2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then (67) can be written in matrix form as (A′) = (G)T(A)(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (68) To prove this equality (and hence (67)) we compute (G)T(A)(G) = � 1 0 zζT z � �γ ℓT ℓ ℓ(2) � � 1 zζ 0 z � = � γ ℓT z (γ(ζ, ·) + ℓ) z � ℓ(ζ) + ℓ(2)� � � 1 zζ 0 z � = � γ z (γ(ζ, ·) + ℓ)T z (γ(ζ, ·) + ℓ) z � γ(ζ, ζ) + 2ℓ(ζ) + ℓ(2)� � , which is precisely the matrix form of A′ after taken into account the transformation laws (8)-(10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Item (2) of Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='1 can be also written in matrix form as (Ψ)T(A)(Ψ) = (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 19 To compute the gauge behaviour of Ψ we impose (Ψ′)T(A′)(Ψ′) = (A′) and use equation (68), (Ψ′)T(A′)(Ψ′) = (A′) ⇐⇒ (Ψ′)T(G)T(A)(G)(Ψ′) = (G)T(A)(G) ⇐⇒ � (G)T�−1 (Ψ′)T(G)T(A)(G)(Ψ′)(G)−1 = (A), from where we conclude that Ψ = G ◦ Ψ′ ◦ G−1 and hence Ψ′ = G−1 ◦ Ψ ◦ G, which is (64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The transformation of µ follows from those of Ψ and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Indeed, by (67), µ′ := A′ (Ψ′(n′, 0), (n′, 0)) = A ((G ◦ Ψ′) (n′, 0), G(n′, 0)) , and using (64) together with G ((n′, 0)) = z−1(n, 0) (and its underlined version), µ′ = A ((Ψ ◦ G) (n′, 0), G(n′, 0)) = A (Ψ (G(n′, 0)) , G(n′, 0)) = z−1z−1A (Ψ ((n, 0)) , (n, 0)) = z−1z−1µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The transformation law of a normal pair in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='5 can be rewritten in terms of the map G defined in (65) by t′ i = G−1 (ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Next we want to define two null and transverse hypersurfaces from an abstract point of view, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=', without seeing them as embedded in any ambient spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In [13] the notion of double embedded CHD was introduced in order to study how two different CHD fit together in the same spacetime when we identify their boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' However, using the notion of normal pair and the map Ψ, it is no longer necessary to introduce such object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Indeed, let D and D be embedded CHD with respective embeddings Φ and Φ in a spacetime (M, g), and suppose Φ(S) = Φ(S) =: S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let φ : S −→ S be the induced diffeomorphism and let Ψ be a ∂-isometry satisfying A (Ψ(n, 0), (n, 0)) = µ ∈ F(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' By Defs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='2, the object A (Ψ(n, 0), (n, 0)) can be thought as the scalar product g(ν, ν) at S, where ν = Φ⋆n and ν = Φ⋆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Thus, if one wants to define abstractly two null and transverse hypersurfaces with the same orientation for ν and ν, the function A (Ψ(n, 0), (n, 0)) must be everywhere negative when n and n point into the interior of H and H, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then by Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='2 the condition A (Ψ(n, 0), (n, 0)) = µ ̸= 0 fixes uniquely the map Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Moreover, if we want Φ(S) and Φ(S) to correspond to the same (codimension two) surface in the ambient spacetime, there are several necessary conditions that they need to fulfil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Firstly, their induced metrics have to agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Secondly, their second fundamental forms and torsion one-forms also need to agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' And finally, the pullback of the ambient Ricci tensor into the two surfaces must agree too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The “zeroth order” condition, namely that the induced metrics coincide, can be simply written as φ⋆h = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In order to write the “first order” conditions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' the ones of the second fundamental forms and torsion one-forms, we can employ the tools developed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' As seen in Section 3, these conditions can be expressed abstractly thanks to the notion of normal pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Indeed, identifying the ambient vectors associated to a basis of normal pairs {ti} with the ambient vectors associated to the normal pairs {Ψ(ti)}, the second fundamental forms Kti and the torsions ℸ(ti, tj) of S must agree with those of S, namely KΨ(ti) and ℸ(Ψ(ti), Ψ(tj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Finally, the “second order” condition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' the one involving the ambient Ricci tensor, can be expressed abstractly thanks to the abstract tensor R defined in (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Indeed, since Φ⋆ Ric = R and Φ⋆ Ric = R (see (35)), the pullback of the tensors R and R on S and S must also agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' This whole discussion motivates the following abstract and fully gauge-covariant definition of double null data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 20 Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let H and H be two manifolds with boundaries i : S −→ H and i : S −→ H, let φ : S −→ S be a diffeomorphism and µ ∈ F(S) everywhere negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let D = {H, γ, ℓ, ℓ(2), Y} and D = {H, γ, ℓ, ℓ(2), Y} be CHD and restrict n and n to point towards the interior of H and H, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let Ψ be the unique ∂-isometry such that A (Ψ(n, 0), (n, 0)) = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let {ti} be a basis of normal pairs of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then we say that the triple {D, D, µ} is double null data (DND) provided that the following conditions hold at S φ⋆h = h, (69) Ψ⋆ � KΨ(ti)� = Kti, (70) Ψ⋆ (ℸ(Ψ(ti), Ψ(tj))) = ℸ(ti, tj), (71) φ⋆ (i⋆R) = i⋆R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (72) In the next remark we show that Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='6 is well-defined, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=', that the compatibility conditions are independent of the basis of NPs and also gauge invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Conditions φ⋆h = h and φ⋆ (i⋆R) = i⋆R are automatically gauge invari- ant by virtue of the transformations laws (8) and (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then it suffices to show the gauge invariance of (70)-(71).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We start with (70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let (z, ζ) and (z, ζ) be gauge parameters and denote by t′ i the gauge transformed normal pair of ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Firstly, the RHS of (70) is simply Kt′ i as a direct consequence of the gauge invariance of the second fundamental form in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We need to show that the RHS can also be written with all objects gauge transformed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let G, G be defined as in Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='6 the LHS of (70) can be written as Ψ⋆ � KΨ(ti)� = Ψ⋆ � KG−1(Ψ(ti))� , since the second fundamental form K along a normal pair is gauge invariant and G−1 (Ψ(ti)) is the gauge transformation of the normal pair Ψ(ti) with gauge parameters (z, ζ) (see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then, using G−1 ◦ Ψ = Ψ′ ◦ G−1 (by Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='4), Ψ⋆ � KG−1(Ψ(ti))� = Ψ⋆ � KΨ′(G−1(ti))� = Ψ⋆ � K(Ψ′(t′ i))� , where in the last equality we used again t′ i = G−1ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Noting that Ψ ((X, 0)) = Ψ′ ((X, 0)) for every X ∈ X(S), the LHS of equation (70) finally gets rewritten as Ψ′⋆ � K(Ψ′(t′ i))� , which is exactly the LHS of (70) with all quantities gauge transformed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' This establishes the gauge covariance of conditions (70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Moreover, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='4, (70) are also inde- pendent of the basis of NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The gauge invariance of conditions (71) can be shown in a similar way as in the case of the second fundamental form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Firstly, the RHS of (71) is simply ℸ(t′ i, t′ j), by the gauge invariance of ℸ in Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' To see that the LHS of (71) can be written with all quantities gauge transformed, we use again G−1 ◦ Ψ = Ψ′ ◦ G−1, t′ i = G−1ti and Ψ ((X, 0)) = Ψ′ ((X, 0)), so that Ψ⋆ (ℸ(Ψ(ti), Ψ(tj))) = Ψ⋆ � ℸ(G−1 (Ψ(ti)) , G−1 (Ψ(tj))) � = Ψ′⋆ � ℸ(Ψ′(t′ i), Ψ′(t′ j)) � , 21 where in the first equality we invoked again the gauge invariance of ℸ (Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Thus, the gauge covariance of (71) is also established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Finally, by Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='13 under a change of basis of NPs ˆti = Ωk i tk, the RHS of (71) transforms as ℸ(ˆti,ˆtj) = Ωk i Ωl jℸ(tk, tl) + Ωk i MkldΩl j, whereas the transformation of the LHS is Ψ⋆ � ℸ(Ψ(ˆti), Ψ(ˆtj)) � = Ψ⋆ �� Ωk i ◦ φ−1� � Ωl j ◦ φ−1� ℸ(Ψ(tk), Ψ(tl)) + � Ωk i ◦ φ−1� M kld � Ωl j ◦ φ−1�� = Ωk i Ωl jΨ⋆ (ℸ(Ψ(tk), Ψ(tl))) + Ωk i MkldΩl j, where we used Ψ⋆M kl = Mkl (by item (2) in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='1), Ψ(ˆti) = Ψ � Ωk i tk � = � Ωk i ◦ φ−1� Ψ(tk) and that the pullback commutes with the exterior derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Since both sides in (71) transform in the same way, conditions (71) are invariant under change of basis of NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The compatibility condition (72), namely φ⋆ (i⋆R) = i⋆R, played no essential role in [13] because in that paper we were interested in data satisfying the abstract constraint equations R = 2Λ m − 1γ and R = 2Λ m − 1γ, (73) so (72) was automatically fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' When (73) do not hold (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' when the field equations are not the Einstein vacuum field equations or no equations whatsoever are imposed) adding this condition is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' An explicit case is Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='15 below where we prove that given double null data {D, D, µ}, there always exists a spacetime in which {D, D, µ} can be embedded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' It turns out that without the condition φ⋆ (i⋆R) = i⋆R the result would not be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' DND has gauge freedom on each component linked by the behaviour of µ as described in Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Thus, we put forward the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let {D, D, µ} be DND and z ∈ F ⋆(H), z ∈ F ⋆(H), ζ ∈ X(H) and ζ ∈ X(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The transformed double null data is given by G ({D, D, µ}) := {D′, D′, µ′}, where D′ and D′ are the transformed CHD in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='3 and µ′ := z−1z−1µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (74) This definition guarantees that when {D, D, µ} is double null data, then G ({D, D, µ}) is double null data too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' This is a straightforward consequence of the gauge covariance of the compatibility conditions (see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In order to connect the abstract definition of double null data with the geometric idea of two null and transverse hypersurfaces, it is necessary to extend the notion of embeddedness to the context of double null data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let {D, D, µ} be DND and (M, g) a spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We say that {D, D, µ} is embedded double null data on (M, g) with riggings ξ, ξ and embeddings Φ, Φ, respec- tively, provided that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' D (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' D) is embedded in (M, g) with embedding Φ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Φ) and rigging ξ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' ξ) in the sense of Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='2 and Φ(S) = Φ(S) =: S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' µ(p) = g(ν, ν)|Φ(p) for all p ∈ S, where ν = Φ⋆n and ν = Φ⋆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 22 Definitions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='10 establishes our intended correspondence between embedded double null data and two transverse, null hypersurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Moreover, since the quantity A (Ψ(n, 0), (n, 0)) is assumed to be negative when n and n point into the interior of H and H, respectively, the two hypersurfaces have the same time-orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let us summarize what we have done so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='6 we have introduced a completely abstract object called double null data that captures the idea of two null and transverse hypersurfaces but without seeing them as embedded in any ambient spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' This object must satisfy certain compatibility conditions at the “corner”, which are clearly necessary for any DND to be able to be embedded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' These conditions have been written in a fully gauge-covariant way thanks to the notion of normal pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' They are also independent of the basis of NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In the following Remark we connect the new completely general definition of double null data with the previous one given in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='5 of [13], where the compatibility equations were written in a particular gauge assuming strong conditions at the boundary as well as a very particular basis of normal pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='3 and Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='16, the compatibility conditions (70) can be written in the basis of normal pairs {tn = (n, 0), tθ = (θ, 1)} as µΨ⋆ (Υ + Θ) = χ, (75) µ−1Ψ⋆χ = Υ + Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (76) In addition, using again Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='16 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='14, (71) can be written as ℸ(tθ, tn) = (Ψ⋆ℸ) (Ψ(tθ), Ψ(tn)) = (Ψ⋆ℸ) � µ−1tn, µtθ � = Ψ⋆τ + d(log |µ|), so employing Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='11 it yields τ + Ψ⋆τ = −d(log |µ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (77) Given D and D characteristic hypersurface data, there exists a class of gauges in which ℓ∥ = 0, ℓ(2) = 0 on S and ℓ∥ = 0, ℓ(2) = 0 on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The existence of these class of gauges was established in [13, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='2] where it was also proved that this family of gauges was parametrized by pairs (z, ζ) and (z, ζ) satisfying ζ|S = 0 and ζ|S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Particularizing equations (75), (76) and (77) to this class of gauges, it yields µY(X, Y ) = K(X, Y ), (78) µ−1K(X, Y ) = Y(X, Y ), (79) Π(X, n) + Π(X, n) = −X(log |µ|) (80) for every X, Y ∈ X(S) (here we omit the pushforward φ⋆ for simplicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' These equations are the same as the compatibility conditions (122)-(124) in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We conclude that Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='6 generalizes our previous definition of double null data in a fully general gauge and in any basis of normal pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' So far it is clear that the compatibility conditions (69)-(72) are necessary for a double null data to be embeddable in some spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The rest of this section is devoted to show that these equations are not only necessary but also sufficient, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=', that if (69)-(72) hold, then there exists a spacetime in which the double null data can be embedded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Here we are not interested in solving any spacetime field equations, we simply want to find 23 that there always exists a spacetime where the data can be embedded, with the aim of showing that we have not forgotten any additional restriction on the data that might have been necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The construction of the spacetime will be based on the harmonic gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='6 we have already recalled the result in [13] that guarantees the existence of a harmonic gauge associated to a set of m functionally independent functions on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We now choose functions {xa} = {u, xA} on H and {xa} = {u, xA} on H with the following properties: (i) n(u) ̸= 0, u|S = 0, n(xA) = 0 and {xA} is a local coordinate system on S (ii) n(u) ̸= 0, u|S = 0, n(xA) = 0 and {xA} is a local coordinate system on S (iii) xA ◦ φ = xA \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (81) Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='7 guarantees the existence of a harmonic gauge associated to the functions {u, xA} (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' {u, xA}) on H (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' H) satisfying ℓ∥ = 0, ℓ(2) = 0 on S and ℓ∥ = 0, ℓ(2) = 0 on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Moreover, the residual gauge freedom is parametrized by pairs (z, z) ∈ F ⋆(S) × F ⋆(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' One can exploit this freedom to fix the value of the functions n(u) and n(u) at S and S, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let {D, D, µ} be DND and consider two set of independent functions {u, xA} and {u, xA} satisfying conditions (81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then there exists a unique harmonic gauge w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='t {u, xA} and {u, xA} in D and D, respectively, in which ℓ∥ = 0, ℓ(2) = 0, n(u) = µ on S and ℓ∥ = 0, ℓ(2) = 0, n(u) = µ on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The transformation law of the functions n(u) and n(u) follow from that of n (see (13)), n′(u) = z−1n(u) and n′(u) = z−1n(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Recalling the transformation of µ in (66), namely µ′ = z−1z−1µ, one can choose z = µn(u)−1 and z = µn(u)−1 so that µ′ = n′(u) = n′(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Applying Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='8 it follows that when the DND is written in the unique gauge defined in the previous Lemma, additional relations between the data at the boundary appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let {D, D, µ} be DND written in the harmonic gauge of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='12 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='t {u, xA} and {u, xA}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then for every X ∈ X(S) the following relations hold at S (we omit the pushforward φ⋆ for simplicity) 2Y(n, n) = µn � ℓ(2)� , (82) 2Y(n, n) = µn � ℓ(2)� , (83) 2Π(X, n) = (Lnℓ) (X) − X (log |µ|) − (Lnℓ) (X), (84) 2Π(X, n) = (Lnℓ) (X) − X (log |µ|) − (Lnℓ) (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (85) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Firstly, from equation (39) it follows n � ℓ(2)� = trh Υ on S, which together with (79) and (37) yields (83).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Equation (82) is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Secondly, from (40) and xA ◦ φ = xA, 2 (Lnℓ + Π(·, n)) (gradhxA) = □hxA = □hxA = 2 (Lnℓ + Π(·, n)) (gradhxA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Employing (80), equations (84) and (85) follow at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 24 In the following remark we compute the compatibility condition (72) in the gauge defined in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='12, for which we first need to write the pullback of the tensor R into a section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' This has been computed in [8] in the case of general null hypersurface data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Their result is fully diffeomorphism covariant, the dependence on the tensor Y is explicit and is written without assuming any gauge condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' However, since we only need R in a very specific gauge and to make this article as self-contained as possible, we redo this computation in Appendix A assuming a gauge in which both ℓ(2) and ℓ∥ vanish at the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In this remark we compute the compatibility condition i⋆R = i⋆R on S in the gauge defined in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='12 (we omit the pullback φ⋆ for simplicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='1 of Appendix A we found that the pullback to S of the abstract constraint tensor R as defined in (34) in a gauge in which ℓ∥ = 0 and ℓ(2) = 0 on S is RAB = Rh AB + (2Y(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' n) − trh K) YAB + � n � ℓ(2)� − trh Y � KAB + 4hCDKC(AYB)D + 2∇h (AτB) + 2∇h (A (Lnℓ)B) − 2LnΠ(AB) − 2τAτB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (86) and analogously the pullback of R into S in the same gauge is RAB = Rh AB + (2Y(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' n) − trh K) YAB + � n � ℓ(2)� − trh Y � KAB + 4hCDKC(AYB)D + 2∇h (Aτ B) + 2∇h (A (Lnℓ)B) − 2LnΠ(AB) − 2τ Aτ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (87) From condition (69), namely that h S= h, it follows Rh AB S= Rh AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' From (78)-(79), the terms trh K YAB, trh Y KAB and 4hCDKC(AYB)D from (86) are equal to the terms trh Y KAB, trh K YAB and 4hCDKC(AYB)D from (87), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Finally from condi- tion (77) we can substitute τ in (87) in terms of τ and d log |µ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then, the compatibility condition RAB = RAB in the gauge in which ℓ∥ = ℓ∥ = 0 and ℓ(2) = ℓ(2) = 0 on S and S can be written as (LnΠ)(AB) − (LnΠ)(AB) = � Y(n, n) − 1 2µn � ℓ(2)�� YAB + �1 2n � ℓ(2)� − µ−1Y(n, n) � KAB + 2∇h (AτB) + ∇h (A (Lnℓ)B) − ∇h (A (Lnℓ)B) + ∇h (A∇h B) log |µ| (88) + 2τ(A∇h B) log |µ| + ∇h A log |µ|∇h B log |µ|, We now restrict further the gauge so that we are in the harmonic gauge of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Taking into account (82)-(85) and the fact that Π(X, n) = τ(X) for every X ∈ X(S), the first and second lines of the RHS of (88) vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Replacing the term 2τA of the third line by (Lnℓ)A − ∇h A log |µ| − (Lnℓ)A (see (84)), equation (88) finally reads (LnΠ)(AB) − (LnΠ)(AB) = (Lnℓ)(A ∇h B) log |µ| − (Lnℓ)(A ∇h B) log |µ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (89) We now have the necessary ingredients to show that (69)-(72) is everything one needs to make sure that double null data can be embedded in some spacetime, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=', that the definition is complete and we are not missing any extra conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let {D, D, µ} be double null data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then there exists a spacetime (M, g), embeddings Φ : H ֒→ M, Φ : H ֒→ M and vector fields ξ, ξ along Φ(H) and Φ(H), respectively, such that {D, D, µ} is embedded double null data in (M, g) with embeddings Φ, Φ and riggings ξ, ξ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let {D, D, µ} be double null data, D = {H, γ, ℓ, ℓ(2), Y} and D = {H, γ, ℓ, ℓ(2), Y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Consider a set of independent functions {u, xA} and {u, xA} satisfying the conditions of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='12, namely (81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Henceforth the coordinates u and u will be assumed to be ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We write the data in the gauge of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='12 with respect to these functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Consider the manifold R × R × S and use u and u as the natural coordinates in the first and second factors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We shall work in the manifold with boundary M := {u, u ≥ 0} ⊂ R2 × S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Any (local) coordinate system {xA} in S extends to a (local) coordinate system {u, u, xA} in M such that {u, xA} (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' {u, xA}) restricted to H (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' H) are the given coordinates on H (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' H), as well as u|H = 0 and u|H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We define the embeddings Φ : H −→ M Φ : H −→ M (u, xA) �−→ (u = 0, u, xA) (u, xA) �−→ (u, u = 0, xA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (90) Thus, Φ(H) = {u = 0} and Φ(H) = {u = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We denote by S the intersection of Φ(H) and Φ(H), namely S := {u = u = 0} ⊂ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let ξ and ξ be defined by ξ := ∂u and ξ := ∂u in the coordinate system we have introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In these coordinates we also have n = λ∂u and n = λ∂u, where λ := n(u) and λ := n(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In order to prove the theorem we only need to construct a smooth metric g on M inducing the given data on H ∪ H (we do not write Φ or Φ for simplicity) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='t the riggings ξ and ξ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=', gu u|H = 0, gu u|H = ℓ(2), gu u|H = ℓ(2), gu u|H = 0, gu A|H = 0, gu A|H = ℓA, gu A|H = ℓA, gu A|H = 0, gAB|H = γAB, gAB|H = γAB, gu u|H = λ−1, gu u|H = λ−1, (91) gu u|S = µ−1, (92) and 1 2 (Lξg)ab = Yab, 1 2 � Lξg � ab = Yab, (93) where ℓA := ℓ(∂xA), γAB := γ(∂xA, ∂xB) and similarly on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The conditions of the first line of (91) follow from the fact that γ(n, ·) = 0 and Φ⋆ (g(ξ, ξ)) = ℓ(2) (see (6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The second line also follows from γ(n, ·) = 0 as well as from Φ⋆ (g(ξ, ·)) = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The third line fol- lows directly from Φ⋆g = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The fourth line follows from 1 = g(ξ, ν) = g(∂u, λ∂u) = λgu u and its underlined version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' These four lines constitute the metric part of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Con- dition (92) follows from µ = g(ν, ν)|S = λλgu u and the fact that λ = λ = µ on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Finally, conditions (93) guarantee that the metric g also induces the Y tensor (see (7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In order to construct such g, our strategy is to extend the components of the hypersurface data tensors in these coordinates to all M and define the components of the metric g in such a way that it induces the given data on H ∪ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Thus, we introduce the notation f H to denote the extension of the function f ∈ F(H) off H satisfying ξ (f) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We define f H analogously, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=', by extending the function f ∈ F(H) by means of ξ(f) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Moreover, f S will denote the extension of f ∈ F(S) off S satisfying ξ (f) = ξ (f) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then we define the components of g in this coordinate system as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Component gu u: Let gu u be defined on M by gu u := (ℓ(2))H+2(YH u u−YS u u)u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Since YH u u = YS u u on H we have gu u|H = ℓ(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' From ℓ(2) S= 0 its extension (ℓ(2))H vanishes on H and hence gu u|H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Concerning the transverse derivative, we first note 26 that for any function f ∈ F(H) it holds ∂u(f H) = (∂uf)H and similarly ∂u(f H) = (∂uf)H for any function f ∈ F(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Indeed, ∂u � ∂u(f H) � = ∂u � ∂u(f H) � = 0 and ∂u � (∂uf)H� = 0 by construction, so the function ∂u(f H)−(∂uf)H is constant along each integral curve of ∂u, and since on H ∂u(f H) = (∂uf)H = ∂uf, we conclude that ∂u(f H) = (∂uf)H (and similarly ∂u(f H) = (∂uf)H on M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Now, condition (82) together with n = λ∂u, n = λ∂u and λ = λ on S gives ∂uℓ(2) S= 2Yu u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Therefore, all along H their extensions agree, (∂uℓ(2))H = ∂u � (ℓ(2))H� = 2YS u u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Consequently, 1 2 (Lξg)u u = 1 2∂ugu u = 1 2∂u � (ℓ(2))H� + YH u u − YS u u H= Yu u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Component gu u: Analogously, we define gu u := (ℓ(2))H + 2 � YH u u − YS u u � u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' By symmetry of the construction this also induces the given data on H and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Component gu u: Let gu u be defined on M by gu u := (λ−1)H + � λ−1�H − (µ−1)S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Since µ = λ = λ on S, gu u|H = λ−1 and gu u|H = λ−1 (and they match on S and fulfil condition (92)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Components gu A: Let gu A := ℓH A + 2(YH u A − YS u A)u on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Since YH u A = YS u A on H we have gu A|H = ℓA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' From ℓA = 0 on S, its extension ℓH A also vanishes on H, and then gu A|H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In order to see that these gu A induce the corresponding components of the Y tensor we start by writing equation (84) in the coordinate system {u, u, xA}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=', taking X = ∂xA, n = λ∂u and n = λ∂u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Using µ S= λ, 2λΠA u S= λ∂uℓA + ℓ(∂u)∂xA(λ) − ∂xA log |λ| − λ∂uℓA − ℓ(∂u)∂xA(λ) S= λ∂uℓA + λ−1∂xA(λ) − ∂xA log |λ| − λ∂uℓA − λ−1∂xA(λ) S= λ∂uℓA − ∂xA log |λ| − λ∂uℓA, (94) where in the first line we used the well-known formula LfXω = fLXω + ω(X)df valid for any function f, vector X and one-form ω, in the second line we used ℓ(n) = λℓ(∂u) = 1 and thus ℓ(∂u) = λ−1 (and its underlined version) and in the third line λ∂xA(λ) S= λ∂xA(λ) since λ = λ on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The value of ΠA u can be computed from (23) and (17), namely 2ΠA u = 2YA u + 2FA u = 2YA u + ∂xAλ−1 − ∂uℓA, (95) where again we used ℓu = λ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Inserting (95) evaluated at S into (94) and taking again into account that λ = λ on S, it yields 2Yu A = ∂uℓA on S and therefore 2YS u A = (∂uℓA)H = ∂u � ℓH A � on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Hence, 1 2 (Lξg)u A = 1 2∂ugu A = 1 2∂u � ℓH A � + YH u A − YS u A H= Yu A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Components gu A: Analogously, we define gu A := ℓH A + 2(YH u A − YS u A)u on M, which by symmetry also induces the given data on H and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Components gAB: Let h be the induced metric on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We define the functions gAB on M by means of gAB := γH AB + γH AB − hS AB + 2 � YH AB − YS AB � u + 2 � YH AB − YS AB � u − 2(∂uYAB)Suu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 27 Since on H γH AB = hS AB and YH AB = YS AB, and on H γH AB = hS AB and YH AB = YS AB, we have gAB|H = γAB and gAB|H = γAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Moreover, from (18) and n = λ∂u, we have ∂uγAB = 2λ−1KAB on H, so in particular ∂uγAB = 2λ−1KAB on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Using λ = µ and µ−1KAB = YAB on S (see (79)) it follows that (∂uγAB)H = ∂u � γH AB � = 2YS AB on H, and thus 1 2 (Lξg)AB = 1 2∂ugAB = 1 2∂u(γH AB) + YH AB − YS AB + ∂u(YH AB)u − (∂uYAB)S u H= 1 2∂u(γAB)H + YH AB − YS AB H= YAB, where in the third equality we used ∂u � YH AB � = (∂uYAB)S on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Before computing Lξg on H we need to write equation (89) in the coordinate system {u, u, xA}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Since [n, ∂xA] = [λ∂u, ∂xA] = −∂xAλ ∂u and λ = µ on S, (LnΠ)AB S= n (ΠAB) + ∂xA(log |µ|)Π (n, ∂xB) + ∂xB(log |µ|)Π (∂xA, n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Using (24) and the fact that F is antisymmetric, (LnΠ)AB + (LnΠ)BA S= 2n (YAB) + (2Π(∂xB, n) + (Lnℓ) (∂xB)) ∂xA(log |µ|) + (2Π(∂xA, n) + (Lnℓ) (∂xA)) ∂xB(log |µ|) S= 2λ∂u (YAB) + (Lnℓ) (∂xB)∂xA(log |µ|) + (Lnℓ) (∂xA)∂xB(log |µ|) − 2∂xA(log |µ|)∂xB(log |µ|), where in the second equality we used (84).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' equation (89) in this coordinate system becomes simply ∂u (YAB) S= ∂u (YAB) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (96) and then the quantity Lξg on H is finally given by 1 2 � Lξg � AB = 1 2∂ugAB = 1 2∂u � γH AB � + ∂u � YH AB � u + YH AB − YS AB − (∂uYAB)S u H= 1 2∂u (γAB)H + YH AB − YS AB H= YAB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' where in the third line we used that on H ∂u � YH AB � = (∂uYAB)S = (∂uYAB)S (the second equality following from (96)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' and in the fourth line that (∂uγAB)H = ∂u � γH AB � = 2YS AB on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Given that gµν fulfils all conditions (91)-(93), we conclude that {D, D, µ} is embedded double null data in (M, g) with embeddings Φ, Φ and riggings ξ, ξ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 28 The previous Theorem shows that any double null data can be embedded in some space- time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' It then arises the natural question of whether it can be also embedded in a spacetime solution of the Einstein equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' By (35), if {D, D, µ} is embedded DND on an (m + 1)-dimensional spacetime (M, g) solution of the Λ-vacuum equations, namely Ric = 2Λ m − 1g, then it must satisfy R = 2Λ m − 1γ and R = 2Λ m − 1γ, where R is the abstract tensor defined in (34) (and analogously for R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Thus, these restrictions are necessary for {D, D, µ} to be embedded in a spacetime solution of the Λ-vacuum Einstein equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The main result in [13] proves that they are also sufficient, as we summarize next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let {D, D, µ} be double null data of dimension m > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We say that a Lorentzian manifold (M, g) is a development of {D, D, µ} provided there exist embeddings Φ, Φ and riggings ξ, ξ such that {D, D, µ} is embedded DND in (M, g) with embeddings Φ, Φ and riggings ξ, ξ in the sense of Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='10 and Φ(H) ∪ Φ(H) = ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' With this definition we can restate Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='15 of [13] in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let {D, D, µ} be double null data of dimension m > 1 as defined in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='6 satisfying the abstract constraint equations R = 2Λ m − 1γ and R = 2Λ m − 1γ, (97) where R is defined in (34), R is its underlined version and Λ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then there ex- ists a development (M, g) of {D, D, µ} (possibly restricted if necessary) solution of the Λ-vacuum Einstein equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Moreover, for any two such developments (M, g) and ( � M, �g), there exist neighbourhoods of H ∪ H, U ⊆ M and �U ⊆ � M, and a diffeomor- phism ϕ : U −→ �U such that ϕ⋆�g = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='15 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='17 establish a very clear hierarchy between the compatibility conditions and the constraint equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The former are the necessary and sufficient conditions for a DND to be able to be embedded in some spacetime, whereas the later are necessary and sufficient for the DND to be embedded in a spacetime solution of the Einstein field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 5 Isometry between Double Null Data Given two double null data, there arises the natural question of under which conditions their developments are the same (up to isometry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In this section we establish the neces- sary and sufficient conditions for two double null data to define two isometric spacetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We start with a definition to fix some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let D = {H, γ, ℓ, ℓ(2), Y} be hypersurface data and ψ : � H −→ H a diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We define the pull-back hypersurface data ψ⋆D by ψ⋆D := � � H, �γ := ψ⋆γ, �ℓ := ψ⋆ℓ, �ℓ(2) := ψ⋆ℓ(2), �Y := ψ⋆Y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 29 From Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='1 and the fact that ψ is a diffeomorphism, it follows that ψ⋆D is still hypersurface data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Moreover, from (1)-(5) having a unique solution for {P, n, n(2)} given {γ, ℓ, ℓ(2)}, it follows that �P = ψ⋆P, �n = ψ⋆n and �n(2) = ψ⋆n(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Thus, the causal character of D is the same as the one of ψ⋆D, and in particular if D is null, so it is ψ⋆D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The previous definition can be extended to the context of double null data as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let {D, D, µ} be double null data and ψ : � H −→ H, ψ : � H −→ H diffeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The pull-back double null data Ξ⋆{D, D, µ} is defined as Ξ⋆{D, D, µ} := � ψ⋆D, ψ⋆D, ψ|⋆ S(µ) � , where ψ⋆D and ψ⋆D are the pull-backs in the sense of Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Since ψ and ψ are diffeomorphisms, they preserve the boundaries S and S, and thus the map �φ := ψ◦φ◦ψ−1 : �S −→ �S is a diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then, Ξ⋆{D, D, µ} is still double null data, since it satisfies Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='6 with �φ and �µ := ψ|⋆ S(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In the following proposition we find the necessary conditions for two DND to define two isometric Lorentzian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let {D, D, µ} and { �D, �D, �µ} be double null data satisfying the con- straint equations (97) and let (M, g) and ( � M, �g) be respective developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Suppose that there exists an isometry ϕ : M −→ � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then there exist gauge parameters (z, ζ) and (z, ζ) in D and D, respectively, and a map Ξ⋆ as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='2 such that Ξ⋆{ �D, �D, �µ} = G ({D, D, µ}) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let �xµ = {�u, �u, �xA} be coordinates on � M whose restrictions on H and H satisfy (81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Define the coordinates xµ on M by xµ := �xµ ◦ ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let Φ, Φ and ξ, ξ the embeddings and the riggings of {D, D, µ} in (M, g), and �Φ, �Φ and �ξ, �ξ be the embeddings and the riggings of { �D, �D, �µ} in ( � M, �g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Since (ϕ⋆�ξ)(u) = �ξ(u ◦ ϕ−1) = �ξ(�u) ̸= 0, there exist z ∈ F ⋆(H) and ζ ∈ X(H) such that ϕ⋆�ξ = z(ξ + Φ⋆ζ) along Φ(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Since Φ(H) is diffeo- morphic to �Φ( � H) via ϕ and both Φ and �Φ are embeddings, there exists a diffeomorphism ψ making the following diagram commutative H � H M � M ψ Φ �Φ ϕ Then ψ⋆�γ = ψ⋆�Φ⋆�g = Φ⋆ϕ⋆�g = Φ⋆g = γ, and ψ⋆�ℓ = ψ⋆�Φ⋆� �g(�ξ, ·) � = Φ⋆ϕ⋆� �g(�ξ, ·) � = Φ⋆ (g(z(ξ + Φ⋆ζ), ·)) = z(ℓ + γ(ζ, ·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Concerning �ℓ(2), ψ⋆�ℓ(2) = ψ⋆�Φ⋆� �g(�ξ, �ξ) � = Φ⋆ � z2g(ξ, ξ) + 2z2g(ξ, Φ⋆ζ) + z2g(Φ⋆ζ, Φ⋆ζ) � = z2(ℓ(2) + 2ℓ(ζ) + γ(ζ, ζ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 30 Finally, ψ⋆ �Y = 1 2ψ⋆�Φ⋆L�ξ�g = 1 2Φ⋆ϕ⋆L�ξ�g = 1 2Φ⋆ � Lz(ξ+Φ⋆ζ)g � = zY + dz ⊗s ℓ + 1 2Lzζγ, where the third equality holds because ϕ⋆�ξ = z (ξ + Φ⋆ζ), ϕ⋆�g = g and3 ϕ⋆� L�ξ�g � = Lϕ⋆�ξ � ϕ⋆g � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Thus, recalling Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='2, ψ⋆ �D = G(z,ζ)(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The same argument on H proves that there exist gauge parameters (z, ζ) on D such that ψ⋆ �D = G(z,ζ)(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Finally, taking into account item 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' of Def 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='10, ψ|⋆ S(�µ) = Φ|⋆ Sϕ⋆ (�g(�ν, �ν)) = Φ|⋆ S � g(z−1ν, z−1ν) � = z−1z−1µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Comparing with (66), the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The previous proposition motivates defining the notion of isometric double null data as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We say that two double null data {D, D, µ} and { �D, �D, �µ} are isometric if there exist diffeomorphisms ψ : H −→ � H and ψ : H −→ �H and gauge parameters (z, ζ) and (z, ζ) in D and D, respectively, such that the pull-back double null data Ξ⋆{ �D, �D, �µ} satisfies Ξ⋆{ �D, �D, �µ} = G ({D, D, µ}) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We conclude this paper by proving that the necessary conditions of Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='3 are also suf- ficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' This result gives a geometric uniqueness statement of the characteristic problem of the Einstein field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Indeed, two isometric initial data are indistinguishable from a geometric point of view and thus they should have “the same” developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The precise statement of the notion of uniqueness is given in the following Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let {D, D, µ}, { �D, �D, �µ} be two isometric DND in the sense of Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='4 satisfying the abstract constraint equations (97), and let (M, g), ( � M, �g) be respective developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then there exist neighbourhoods U ⊆ M and �U ⊆ � M of H ∪ H and � H ∪ � H, respectively, and a diffeomorphism ϕ : U −→ �U such that ϕ⋆�g = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We start by writing { �D, �D, �µ} in the gauge of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='12 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='t some coordi- nates {�xa} = {�u, �xA} and {�xa} = {�u, �xA} in H and H satisfying (81), and {D, D, µ} in the gauge in which Ξ⋆{ �D, �D, �µ} = {D, D, µ} holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We want to show that Ξ⋆{ �D, �D, �µ} is written in the gauge of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='12 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='t the coordinates {xa} := {�xa ◦ ψ} and {xa} := {�xa ◦ ψ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let �V be the vector field defined in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='6 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='t {�xa}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' First we prove that ψ⋆ �V is again the vector of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='6 but w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='t {ψ⋆�xa} (and anal- ogously on H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let {�ec} be a (local) basis of X( � H) and {ec := ψ⋆�ec} a (local) ba- sis of X(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Since ψ⋆ (�ec(�xa)) = ec(xa), it follows ψ⋆ �Bca = Bca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Moreover, since ψ⋆ �∇ = ∇ (because Ξ⋆{ �D, �D, �µ} = {D, D, µ}), the pull-back of the Hessian of a function is the Hessian of the pullback of that function, and since ψ⋆ �P = P, it turns out that ψ⋆� �P ab �∇a �∇b�xa� = P ab∇a∇bxa, and thus ψ⋆ �V c = ψ⋆� �Bca �□ � P �xa� = Bca□Pxa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' There- fore Ξ⋆{ �D, �D, �µ} is written in a harmonic gauge w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='t {xa} and {xa}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Moreover, since 3This follows from the known formula ϕ⋆ (Lϕ⋆XT ) = LX (ϕ⋆T ) valid for any diffeomorphism ϕ, vector X and tensor T particularized to T = �g and ϕ⋆X = �ξ =⇒ X = ϕ⋆�ξ = z (ξ + Φ⋆ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 31 PSfrag replacements (M, g) ( � M, �g) ϕ U �U Figure 1: Given isometric DND, there exist isometric neighbourhoods U and �U of the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' ψ⋆�ℓ(2) = ℓ(2) = 0, ψ⋆�ℓ∥ = ℓ∥ = 0 on S, ψ⋆�ℓ(2) = ℓ(2) = 0, ψ⋆�ℓ∥ = ℓ∥ = 0 on S, as well as ψ⋆ S(�µ) = ψ⋆ S (�n(�u)) = � ψ⋆�n � (u) on S and similarly ψ|⋆ S(�µ) = (ψ⋆�n) (u) on S (we omit the φ⋆ for simplicity), we conclude that the data Ξ⋆{ �D, �D, �µ} is written in the gauge of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='12 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='t the coordinates {xa} and {xa}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let (M, g) be a development of {D, D, µ} with embeddings Φ, Φ and riggings ξ, ξ and let ( � M, �g) be the same but everything with a “�”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let {xµ} be the harmonic coordinates on (M, g) restricting to the given ones at H ∪ H and satisfying Φ(H) = {u = 0}, Φ(H) = {u = 0} (and the same with “�”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' It is straightforward to see that the rigging vectors are given by ξ = ∂u, ξ = ∂u and �ξ = ∂�u, �ξ = ∂�u (we refer the reader to the proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='15 of [13] for the details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Choosing suitable neighbourhoods U ⊆ M of H ∪ H and �U ⊆ � M of � H ∪ �H (see figure 1), we can define the diffeomorphism ϕ by xµ = �xµ ◦ ϕ, which by construction restricts to the given diffeomorphisms ψ : H −→ � H and ψ : H −→ �H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Since ( � M, �g) is a solution of the EFE with cosmological constant Λ, so it is (M, ϕ⋆�g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Moreover, □ϕ⋆�gxµ = 0, so both g and ϕ⋆�g are a solution of the reduced equations in the coordinates {xµ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' By theorem 1 of [15], in order to prove that g = ϕ⋆�g on U, we only need to show that their restrictions to H ∪ H agree, which follows at once after recalling Ξ⋆{ �D, �D, �µ} = {D, D, µ} and using that ϕ restricts to ψ and ψ on H and H, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' A Abstract constraint tensor in a section Let D = {H, γ, ℓ, ℓ(2), Y} be null hypersurface data admitting a section i : S ֒→ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' As shown in [13, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='4], one can always choose a gauge in which ℓ∥ = 0 and ℓ(2) = 0 on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In this appendix we compute the pullback of the abstract constraint tensor R as defined in (34) into S in the aforementioned gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let D = {H, γ, ℓ, ℓ(2), Y} be null hypersurface data admitting a section i : S ֒→ H and let R be the abstract constraint tensor (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then the pullback of R into S in a gauge in which ℓ(2) = 0 and ℓ∥ = 0 on S takes the following form RAB = Rh AB + (2Y(n, n) − trh K) YAB + � n � ℓ(2)� − trh Y � KAB + 4hCDKC(AYB)D + 2∇h (AτB) + 2∇h (A (Lnℓ)B) − 2 (LnΠ)(AB) − 2τAτB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (98) 32 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Let A and B be the tensors defined by Abca := ℓdR d bca + 2ℓ(2)∇[aKc]b + Kb[c∇a]ℓ(2), (99) Babcd := γafR f bcd + 2∇[d � Kc]bℓa � + 2ℓ(2)Kb[cKd]a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (100) Comparing (34) with (99)-(100), the tensor R can be written as Rab = BacbdP cd + (Abca + Aacb) nc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (101) Let {eA} be a (local) basis of TS with dual basis {θA}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=', θA(eB) = δA B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then {n, eA} is a (local) basis of TS ⊕ span{n}, and since ℓ∥ = 0, {ℓ, θA} is the dual basis of {n, eA}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' By equations (3)-(4), the tensor P takes the following form in the basis {n, eA} P = hABeA ⊗ eB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (102) In order to write down the tensor RAB := Rabea Aeb B we divide the computation into two parts, namely BacbdP cdea Aeb B and Abcancea Aeb B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' To calculate the former we need to recall some results concerning the relation between ∇ and the Levi-Civita connection ∇h of h := i⋆γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' As a consequence of Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='5 and equation (30) of [13] together with ℓ∥ = 0 and ℓ(2) = 0, the relation between ∇ and ∇h is ∇XY = ∇ h XY − Y(X, Y )n, X, Y ∈ X(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (103) As usual, this decomposition allows us to relate the completely tangential components of the curvature tensor of ∇ with the curvature tensor of ∇h via a Gauss-type identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The explicit expression is obtained in [13, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='7] and in the present gauge reads γ � W, R(X, Y )Z � = h � W, Rh(X, Y )Z � − Y(Y, Z)K(X, W) + Y(X, Z)K(Y, W), (104) where R and Rh are the curvature tensors of ∇ and ∇h, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We now have all the ingredients needed to compute BacbdP cdea Aeb B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Taking into account that ℓ(2) = 0 and using (102), BacbdP cdea Aeb B = � γcfR f adb + 2∇[b � Kd]aℓc �� hCDec Ced Dea Aeb B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (105) For the first term we employ Gauss identity (104) and equation (102), γcfR f adbhCDec Ced Dea Aeb B = γ � eC, R(eD, eB)eA � hCD = Rh AB − YAB trh K + hCDYDAKBC, where Rh AB is the Ricci tensor of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In this appendix, when no confusion arises we denote with the same symbol a tensor on H and its pullback on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' For the second term of (105) we use (21), which in this gauge is ∇aℓb S= Πab, and the fact that ΠAB S= YAB (because ℓ∥ = 0 and thus 2i⋆F = i⋆dℓ = dℓ∥ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Then, 2hCDec Ced Dea Aeb B∇[b � Kd]aℓc � = 2hCDec Ced Dea Aeb BKa[d∇b]ℓc = −2hCDKA[BYD]C, where the first equality holds because ℓ∥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Combining the two terms, (105) finally reads BacbdhCDec Ced Dea Aeb B = Rh AB − YAB trh K − KAB trh Y + 2hCDYC(AKB)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (106) 33 Next we compute the terms of the form Abcancea Aeb B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' In the present gauge the quantity Abcanc is simply (note that ℓ(2) is not assumed to be zero off S) nc � ℓdR d bca + Kb[c∇a]ℓ(2)� = ncℓdR d bca − 1 2Kban � ℓ(2)� , (107) where K(n, ·) = 0 has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' The first term in (107) can be computed directly from the Ricci identity, ncℓdR d bca = 2nc∇[a∇c]ℓb = 2nc � ∇[aΠc]b − Kb[c∇a]ℓ(2)� = nc∇aΠcb − ∇nΠab + Kabn � ℓ(2)� , (108) where we used ∇aℓb = Πab − ℓ(2)Kab and that K(n, ·) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Contracting the first term in (108) with ea Aeb B and using (27) and (24) ea Aeb Bnc∇aΠcb = ea Aeb B∇a (Πcbnc) − ea Aeb BΠcb∇anc = ea Aeb B∇a (Πbcnc + Lnℓb) − ea Aeb BΠcb � P cdKda − Πadncnd� = eA (Π(eB, n) + (Lnℓ)(eB)) − (Π(·, n) + Lnℓ) � ∇eAeb B � − hCDec Ced Dea Aeb BΠcbKda + ea Aeb B (Πbc + Lnℓb) Πadncnd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Introducing (103) and recalling Π(n, n) = Y(n, n), ΠAB = YAB and the fact that in this gauge Π(eA, n) = τA (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content='5) this expression finally yields ea Aeb Bnc∇aΠcb = � ∇h eA (τ + Lnℓ) � (eB) + Y(n, n)YAB − hCDYCBKDA + τAτB + τA (Lnℓ)B , (109) where we also used (Lnℓ) (n) = Ln (ℓ(n)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Equation (27) can be rewritten as ∇Xn = K♯(X) − Π(X, n)n, (110) where K♯ is the endomorphism defined by K♯(X) := P (K(X, ·), ·), or in abstract in- dex notation (K♯)ab = P acKcb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' Using ∇nX = ∇Xn + LnX together with (110), the contraction of the second term in (108) with tangential directions can be written as XaY b∇nΠab = n (Π(X, Y )) − Π � ∇nX, Y � − Π � X, ∇nY � = Ln (Π(X, Y )) − Π � ∇Xn + LnX, Y � − Π � X, ∇Y n + LnY � = (LnΠ) (X, Y ) − Π � K♯(X) − Π(X, n)n, Y � − Π � X, K♯(Y ) − Π(Y, n)n � , and therefore ea Aeb B∇nΠab = (LnΠ)AB − hCDYCBKDA + 2τAτB + τA (Lnℓ)B − hCDYACKDB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' (111) Contracting (108) with ea Aeb B and introducing (109) and (111), ea Aeb BncℓdR d bca = � ∇h eA (τ + Lnℓ) � (eB) − (LnΠ)AB + Y(n, n)YAB + hCDYACKDB − τAτB + n � ℓ(2)� KAB, and thus (Aacb + Abca) ncea Aeb B = 2∇h (AτB) + 2∇h (A (Lnℓ)B) − 2 (LnΠ)(AB) + 2Y(n, n)YAB + 2hCDKC(AYB)D − 2τAτB + n � ℓ(2)� KAB, (112) Finally, combining (106) and (112), (98) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 34 Acknowledgements This work has been supported by Projects PID2021-122938NB-I00 (Spanish Ministe- rio de Ciencia e Innovaci´on and FEDER “A way of making Europe”) and SA096P20 (JCyL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' S´anchez-P´erez also acknowledges support of the PhD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' grant FPU20/03751 from Spanish Ministerio de Universidades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' We are very grateful to Miguel Manzano for 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' University of Chicago Press, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} +page_content=' 36' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE0T4oBgHgl3EQf3QI4/content/2301.02722v1.pdf'} diff --git a/WtE3T4oBgHgl3EQf1AtK/content/tmp_files/2301.04742v1.pdf.txt b/WtE3T4oBgHgl3EQf1AtK/content/tmp_files/2301.04742v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..253c2dddb0a50d9d62538f6e99d66d40ae749348 --- /dev/null +++ b/WtE3T4oBgHgl3EQf1AtK/content/tmp_files/2301.04742v1.pdf.txt @@ -0,0 +1,890 @@ +HADA: A Graph-based Amalgamation +Framework in Image-text Retrieval +Manh-Duy Nguyen1[0000−0001−6878−7039], Binh T. Nguyen2,3,4, and Cathal +Gurrin1[0000−0003−2903−3968] +1 School of Computing, Dublin City University, Ireland +2 VNU-HCM, University of Science, Vietnam +3 Vietnam National University Ho Chi Minh City, Vietnam +4 AISIA Lab, Vietnam +Abstract. Many models have been proposed for vision and language +tasks, especially the image-text retrieval task. All state-of-the-art (SOTA) +models in this challenge contained hundreds of millions of parameters. +They also were pretrained on a large external dataset that has been +proven to make a big improvement in overall performance. It is not easy +to propose a new model with a novel architecture and intensively train +it on a massive dataset with many GPUs to surpass many SOTA mod- +els, which are already available to use on the Internet. In this paper, we +proposed a compact graph-based framework, named HADA, which can +combine pretrained models to produce a better result, rather than build- +ing from scratch. First, we created a graph structure in which the nodes +were the features extracted from the pretrained models and the edges +connecting them. The graph structure was employed to capture and fuse +the information from every pretrained model with each other. Then a +graph neural network was applied to update the connection between the +nodes to get the representative embedding vector for an image and text. +Finally, we used the cosine similarity to match images with their rele- +vant texts and vice versa to ensure a low inference time. Our experiments +showed that, although HADA contained a tiny number of trainable pa- +rameters, it could increase baseline performance by more than 3.6% in +terms of evaluation metrics in the Flickr30k dataset. Additionally, the +proposed model did not train on any external dataset and did not require +many GPUs but only 1 to train due to its small number of parameters. +The source code is available at https://github.com/m2man/HADA. +Keywords: image-text retrieval · graph neural network · fusion model. +1 +Introduction +Image-text retrieval is one of the most popular challenges in vision and lan- +guage tasks, with many state-of-the-art models (SOTA) recently introduced +[19,3,18,28,10,25,17]. This challenge includes 2 subtasks, which are image-to- +text retrieval and text-to-image retrieval. The former subtask is defined as an +arXiv:2301.04742v1 [cs.CV] 11 Jan 2023 + +2 +Manh-Duy et al. +image query is given to retrieve relevant texts in a multimodal dataset, while +the latter is vice versa. +Most of the SOTA models in this research field shared 2 things in common: +(1) they were built on transformer-based cross-modality attention architectures +[3,19] and (2) they were pretrained on the large-scale multimodal data crawled +from the Internet [28,18,19,17,13]. However, these things have their own disad- +vantages. The attention structure between 2 modalities could achieve an accurate +result, but it cost a large amount of inference time due to the massive compu- +tation. For instance, UNITER [3] contained roughly 303 millions parameters +and it took a decent amount of time to perform the retrieval in real-time [31]. +Many recent work has resolved this model-related problem by introducing joint- +encoding learning methods. They can learn visual and semantic information from +both modalities without using any cross-attention modules, which can be applied +later to rerank the initial result [18,25,31]. Figure 1 illustrated the architecture +of these pipelines. Regarding the data perspective, the large collected data usu- +ally come with noisy annotation, and hence could be harmful to the models that +are trained on it. Several techniques have been proposed to mitigate this issue +[19,18,17]. However, training on the massive dataset still creates a burden on the +computation facility, such as the number of GPUs, which are required to train +the model successfully and efficiently [28]. +Fig. 1. Two most popular pipelines of the SOTA for image-text retrieval challenge. (a) +A cross-modality transformer network is applied to measure the similarity between an +image and a text based on their features. (b) Each modality used their own transformer +network to get its global embedding. +It has motivated us to answer the question: Can we combine many SOTA +models, which are currently available to use, to get a better unified model without +intensively training with many GPUs? In this paper, we introduced a graph- +based amalgamation framework, called HADA, which formed a graph-based +structure to fuse the features produced by other pretrained models. We did not + +Similarity Score +Similarity Score +Linear Layers +Linear +Linear +Layers +Layers +Cross-modality Transformer +ImageTransformer +Text Transformer +ImageFeature +Global ImageFeature +Image Encoder +Text Encoder +Image Encoder +Text Encoder +TextFeature +GlobalTextFeature +Image Input +Text Input +[CLS] Feature +Image Input +Text Input +(a) +(b)HADA in Image-text Retrieval +3 +use any time-consuming cross-modality attention network to ensure fast retrieval +speed. A graph neural network was employed to extract visual and textual em- +bedded vectors from fused graph-based structures of images and texts, where we +can measure their cosine similarity. To the best of our knowledge, the graph struc- +ture has been widely applied in the image-text retrieval challenge [26,7,27,35,21]. +Nevertheless, it was utilized to capture the interaction between objects or align +local and global information within images. HADA is the first approach that +applies this data structure to combine SOTA pretrained models by fusing their +features in each modality. We trained HADA only on the Flickr30k dataset with- +out using any large-scale datasets. We applied Momentum Distillation technique +[18], which was shown that can not only mitigate the harmful effect of noise an- +notation but also improve the accuracy on a clean dataset. Our experiments +showed that HADA, with the tiny extra number of training parameters, could +improve total recalls by 3.64% compared to the input SOTA without training +with millions of additional image-text pairs as other models. This is the most +crucial part since it is not easy to possess multiple GPUs to use, especially for +small and medium businesses or start-up companies. Therefore, we believe that +HADA can be applied not only in the academic field, but also in industry. +Our main contribution can be summarised as follow: (1) We introduced +HADA, a compact pipeline that can combine 2 or many SOTA pretrained mod- +els to address the image-text retrieval challenge. (2) We proposed a way to fuse +the information between input pretrained models by using graph structures. (3) +We evaluated the performance of HADA on the well-known Flickr30k dataset +[37] and MSCOCO dataset [20] without using any other large-scale dataset but +still improved the accuracy compared to the baseline input models. +2 +Related Work +A typical vision-and-language model, including image-text retrieval task, was +built with the usage of transformer-based encoders. In specific, OSCAR [19], +UNITER [3], and VILLA [10] firstly employed Faster-RCNN [29] and BERT [6] +to extract visual and text features from images and texts. These features were +then fed into a cross-modality transformer block to learn the contextualized +embedding that captured the relations between regional features from images +and word pieces from texts. An additional fully connected layer was used to +classify whether the images and texts were relevant to each other or not, based +on the embedding vectors. Although achieving superior results, these approaches +had a drawback of being applied to real-time use cases. It required a huge amount +of time to perform the retrieval online, since the models have to process the +intensive cross-attention transformer architecture many times for every single +query [31]. +Recently, there have been some works proposing an approach to resolve +that problem by utilising 2 distinct encoders for images and text. The data +from each modality now can be embedded offline and hence improve the re- +trieval speed [31,18,17,25,13,28]. In terms of architecture, all approaches used + +4 +Manh-Duy et al. +the similar BERT-based encoder for semantic data but different image encoders. +While LightningDOT [31] encoded images with detected objects extracted by +the Faster-RCNN model, FastnSlow [25] applied the conventional Resnet net- +work to embed images. On the other side, ALBEF [18] and BLIP [17] employed +the Vision Transformer backbone [8] to get the visual features corresponding +to their patches. Because these SOTA did not use the cross-attention struc- +ture, which was a critical point to achieve high accuracy, they applied different +strategies to increase performance. Specifically, pretraining a model on a large +dataset can significantly improve the result [18,19,13]. For instance, CLIP [28] +and ALIGN [13] were pretrained on 400 millions and 1.8 billions image-text pairs, +respectively. Another way was that they ran another cross-modality image-text +retrieval model to rerank the initial output and get a more accurate result [18,31]. +Regarding to graph structures, SGM [35] introduced a visual graph encoder +and a textual graph encoder to capture the interaction between objects appear- +ing in images and between the entities in text. LGSGM [26] proposed a graph +embedding network on top of SGM to learn both local and global information +about the graphs. Similarly, GSMN [21] presented a novel technique to assess the +correspondence of nodes and edges of graphs extracted from images and texts +separately. SGRAF [7] build a reasoning and filtration graph network to refine +and remove irrelevant interactions between objects in both modalities. +Although there are many SOTAs with different approaches for image-text +retrieval problems, there is no work that tries combining these models but intro- +ducing a new architecture and pretrain on a massive dataset instead. Training +an entire new model from scratch on the dataset is not an easy challenge since it +will create a burden on the computation facilities such as GPUs. In this paper, +we introduced a simple method which combined the features extracted from the +pretrained SOTA by applying graph structures. Unlike other methods that also +used this data structure, we employed graphs to fuse the information between +the input features, which was then fed into a conventional graph neural network +to obtain the embedding for each modality. Our HADA consisted of a small +number of trainable parameters, hence can be easily trained on a small dataset +but still obtained higher results compared to the input models. +3 +Methodology +This section will describe how our HADA addressed the retrieval challenge by +combining any available pretrained models. Figure 2 depicted the workflow of +HADA. We started with only 2 models (Nmodels = 2) as illustrated in Figure +2 for simplicity. Nevertheless, HADA can be extended with a larger Nmodels. +HADA began with using some pretrained models to extract the features from +each modality. We then built a graph structure to connect the extracted fea- +tures together, which were fed into a graph neural network (GNN) later to +update them. The outputs of the GNN were concatenated with the original +global features produced by the pretrained models. Finally, simple linear layers +were employed at the end to get the final representation embedding features for + +HADA in Image-text Retrieval +5 +images and texts, which can be used to measure the similarity to perform the +retrieval. For evaluation, we could extract our representation features offline to +guarantee the high speed inference time. +Fig. 2. The pipeline of the proposed HADA. The red borders indicated trainable com- +ponents. The ITM and ITC infered the training tasks which will be discussed later. +3.1 +Revisit State-of-the-art Models +We only used the pretrained models without using the cross-modality trans- +former structure to extract the features as depicted in Figure 1 to reduce the +number of computations and ensure the high speed inference time. Basically, +they used a unimodal encoder to get the features of an image or a text followed +by a transformer network to embed them and obtain the [CLS] embedding. +This [CLS] token was updated by 1 or many fully connected layers to become a +representative global feature that can be compared with that of the remaining +modality to get the similarity score. +HADA began with the output of the transformer layer from the pretrained +models. In detail, for an input image I, we obtained the sequence of patch tokens +from each model i denoted as v(i) = {v(i) +cls, v(i) +1 , v(i) +2 , ..., v(i) +Ni}, where v(i) +j +∈ Rd(i) +v +and Ni was the length of the sequence. This length depended on the architecture +of the image encoder network employed in the pretrained model. For example, it +could be the number of patches if the image encoder was a Vision Transformer +(ViT) network [8], or the number of detected objects or regions of interest if +the encoder was a Faster-RCNN model [29]. Additionally, we also extracted +the global visual representation feature h(i) +v +∈ Rd(i) +h +from v(i) +cls as illustrated in +Figure 1. Regarding the semantic modality, we used the same process as that +of the visual modality. Specifically, we extracted the sequence of patch tokens +w(i) = {w(i) +cls, w(i) +1 , w(i) +2 , ..., w(i) +L } where w(i) +j +∈ Rd(i) +w +and L was the length of + +Pretrained +Model 1 +Creating +LinearLayers +Graph Structure +Graph Neural +Linear +Image Input +Network +Layers +Linear Layers +Pretrained +Model 2 +ITC / ITM +Pretrained +Model 1 +Creating +LinearLayers +Graph Structure +Graph Neural +Linear +Text Input +Network +Layers +Linear Layers +Pretrained +Model 2 +Image Feature +[CLS] Image Feature +Global Image Feature +Final Image Feature +Text Feature +[CLS] Text Feature +Global Text Feature +Final Text Feature6 +Manh-Duy et al. +the text, and the global textual representation embedding h(i) +w +∈ Rd(i) +h +for an +input text T using the pretrained model i. The input model i matched a pair +of an image I and a text T by calculating the dot product ⟨h(i) +v , h(i) +w ⟩ of their +global features. However, HADA not only used the global embedding but also +the intermediate transformer tokens to make the prediction. We used our learned +[CLS] tokens to improve the global features. In contrast, using the original global +features could ensure high performance of the pretrained models and mitigate +the effect of unhelpful tokens. +3.2 +Create Graph Structure +Each pretrained model i produced different [CLS] features v(i) +cls and w(i) +cls for an +image and text, respectively. Since our purpose was to combine the models, we +needed to fuse these [CLS] tokens to obtain the unified ones for each modality +separately. In each modality, for example, the visual modality, HADA not only +updated v(i) +cls based on v(i) solely but also on those of the remaining pretrained +models {v(j) | j ̸= i}. Because these v came from different models, their dimen- +sions could be not similar to each other. Therefore, we applied a list of linear +layers f (i) +v +: Rd(i) +v +→ Rdp to map them in the same dimensional space: +p(i) = {f (i) +v (x)|x ∈ v(i)} = {p(i) +cls, p(i) +1 , p(i) +2 , ..., p(i) +Ni} +We performed a similar process for the textual modality to obtain: +s(i) = {f (i) +w (x)|x ∈ w(i)} = {s(i) +cls, s(i) +1 , s(i) +2 , ..., s(i) +L }, where f (i) +w : Rd(i) +w → Rds +We then used graph structures Gp = {Vp, Ep} and Gs = {Vs, Es} to connect +these mapped features together, where V and E denoted the list of nodes and +edges in the graph G accordingly. In our HADA, nodes indicated the mapped +features. Specifically, Vp = {p(i)} and Vs = {s(i)} for all i ∈ [1, Nmodels]. Re- +garding edges, we symbolized ea→b as a directed edge from node a to node b in +the graph, thus the set of edges of the visual graph Ep and the textual graph Es +were: +Ep = {ex→p(j) +cls | x ∈ p(i) and i, j ∈ [1, Nmodels]} +Es = {ex→s(j) +cls | x ∈ s(i) and i, j ∈ [1, Nmodels]} +To be more detailed, we created directed edges that went from every patch +features to the [CLS] feature, including from the [CLS] itself, for all pretrained +models but not in the reversed direction, as shown in Figure 2. The reason was +that [CLS] was originally introduced as a representation of all input data, so it +would summarize all patch tokens [8,2,6]. Therefore, it would be the node that +received information from other nodes in the graph. This connection structure +ensured that HADA can update the [CLS] tokens based on the patch tokens +from all pretrained models in a fine-grained manner. + +HADA in Image-text Retrieval +7 +3.3 +Graph Neural Network +Graph neural networks (GNN) have witnessed an increase in its popularity +over the past few years, with many GNN structures having been introduced re- +cently [15,5,34,11,30,1]. HADA applied the modified Graph Attention Network +(GATv2), which was recommended to be used as a baseline whenever employing +GNN [1], to fuse the patch features from different pretrained models together +to get the unified [CLS] features. Let Nk = {x ∈ V | ex→k ∈ E} be the set +of neighbor nodes from which there was an edge connecting to node k in the +graph G. GATv2 used a scoring function se to weight every edge indicating the +importance of the neighbor nodes x in Nk before updating the node k ∈ Rd: +se(ex→k) = A⊤LeakyRELU(W1x + W2k]) +where A ∈ Rd′, W1 ∈ Rd′×d, and W2 ∈ Rd′×d were learnable parameters. These +weights were then normalized across all neighbor nodes in Nk by using a softmax +function to get the attention scores: +αex→k = +exp(se(ex→k)) +� +y∈Nk exp(se(ey→k)) +The updated node k′ ∈ Rd′ was then calculated based on its neighbors in Nk, +including k if we add an edge connect it to itself: +k′ = σ( +� +x∈Nk +αex→k · W1x) +where σ was an nonlinearity activate function. Furthermore, this GATv2 network +could be enlarged by applying a multi-head attention structure and improved +performance [34]. The output now was a concatenation of each head output, +which was similar to Transformer architecture [33]. An extra linear layer was +used at the end to convert these concatenated nodes to the desired dimensions. +We used distinct GATv2 structures with H attention heads for each modality +in this stage, as illustrated in Figure 2. HADA took the input graphs Gp and Gs +with nodes Vp and Vs in the vector space of dp and ds dimensions and updated +them to V′p = {p′(i)} and V′s = {s′(i)} with dimensions of d′ +p and d′ +s. We +then concatenated the updated [CLS] nodes p′ +cls and s′ +cls from all pretrained +models with their corresponding original global embedding hv and hw. Finally, +we fed them into a list of linear layers to get our normalized global representation +hp ∈ Rdh and hs ∈ Rdh. +3.4 +Training Tasks +Image-Text Contrastive Learning. HADA encoded the input image I and +text T to hp and hs, accordingly. We used a similarity function that was a dot +product S(I , T) = ⟨hp, hs⟩ = h⊤ +p hs to ensure that a pair of relevant image-text +(positive pair) would have a higher similar representation compared to irrelevant + +8 +Manh-Duy et al. +pairs (negative pairs). The contrastive loss for image-to-text (i2t) retrieval and +text-to-image (t2i) retrieval for the mini-batch of M relevant pairs (I m, T m) +were: +Li2t(I m) = −log +exp(S(I m, T m)/τ) +�M +i=1 exp(S(I m, T i)/τ) +Lt2i(T m) = −log +exp(S(T m, I m)/τ) +�M +i=1 exp(S(T m, I i)/τ) +where τ was a temperature parameter that could be learned during training. +This contrastive learning has been used in many vision-and-language models +and has been proven to be effective [18,31,17,28]. In our experiment, we trained +HADA with the loss that optimized both subtasks: +LIT C = 1 +M +M +� +m=1 +(Li2t(I m) + Lt2i(T m)) +Inspired by ALBEF [18], we also applied momentum contrast (MoCo) [12] +and their momentum distillation strategy for this unsupervised representation +learning to cope with the problem of noisy information in the dataset and im- +prove accuracy. +Image-Text Matching This objective was a binary classification task to dis- +tinguish irrelevant image-text pairs, but were similar representations. This task +would ensure that they were different in fine-grained details. We implemented +an additional disciminator layer dc : R4dh → R on top of the final embedding +features hp and hs to classify whether the image I and the text T is a positive +pair or not: +dc(hp, hs) = sigmoid(W⊤[hp∥hs∥abs(hp − hs)∥hp ⊙ hs]) +where W ∈ R4dh was trainable parameters, ∥ indicated the concatenation, abs(.) +was the absolute value, and ⊙ denoted elementwise multiplication. We used +binary cross-entropy loss for this ordinary classification task: +Litm(I , T) = ylog(dc(hp, ds)) + (1 − y)log(1 − dc(hp, ds)) +where y was the one-hot vector representing the ground truth label of the pair. +For each positive pair in the minibatch of M positive pairs, we sampled 1 +hard negative text for the image and 1 hard negative image for the text. These +negative samples were chosen from the current mini-batch in which they were +not relevant based on the ground-truth labels, but have the highest similarity +dot product score. Therefore, the objective for this task was: +LIT M = +1 +3M +M +� +m=1 +(Litm(I m, T m) + Litm(I m, T ′ +m) + Litm(I ′ +m, T m)) + +HADA in Image-text Retrieval +9 +where T ′ +m and I ′ +m were the hard negative text and image samples in the mini- +batch that were corresponding with the I m and T m, respectively. The final loss +function in HADA was: +L = LIT C + LIT M +4 +Experiment +4.1 +Dataset and Evaluation Metrics +We trained and evaluated HADA on 2 different common datasets in the image- +text retrieval task which are Flickr30k [37] and MSCOCO [20]. Flickr30k dataset +consists of 31K images collected on the Flickr website, while MSCOCO comprises +123K images. Each image contains 5 relevant texts or captions that describe the +image. We used Karpathy’s split [14], which has been widely applied by all mod- +els in the image-text retrieval task, to split each dataset into train/evaluate/test +on 29K/1K/1K and 113K/5K/5K images on Flickr30k and MSCOCO, respec- +tively. +The common evaluation metric in this task is the Recall at K (R@K) when +many SOTAs used this metric [18,31,17,28,13,19,3,10]. This metric is defined +as the proportion of the number of queries that we found the correct relevant +output in the top K of the retrieved ranked list: +R@K = 1 +Nq +Nq +� +q=1 +1(q, K) +where Nq is the number of queries and 1(q, K) is a binary function returning 1 +if the model find the correct answer of the query q in the top K of the retrieved +output. In particular, for the image-to-text subtask, R@K is the percentage of +the number of images where we found relevant texts in the top K of the output +result. In our experiment, we used R@1, R@5, R@10, and RSum which was the +sum of them. +4.2 +Implementation Details +In our experiment, we combined 2 SOTA models that had available pretrained +weights fine-tuned on the Flickr30k dataset: ALBEF5 and LightningDOT6. None +of them used the cross-modality transformer structure when retrieved to ensure +the fast inference speed7. Although they used the same BERT architecture to +encode a text, the former model employed the ViT network to encode an im- +age, while the latter model applied the Faster-RCNN model. We chose these 2 +5 https://github.com/salesforce/ALBEF +6 https://github.com/intersun/LightningDOT +7 Indeed, these 2 models applied the cross-modality transformer network to rerank the +initial result in the subsequent step. However, we did not focus on this stage. + +10 +Manh-Duy et al. +models because we wanted to combine different models with distinct embedding +backbones to utilize the advantages of each of them. +Regarding ALBEF, their ViT network encoded an image to 577 patch to- +kens including the [CLS] one (NALB = 576 and d(ALB) +v += 768). This [CLS] was +projected to the lower dimension to obtain the global feature (d(ALB) +h += 256). +Because LightningDOT encoded an image based on the detected objects pro- +duced by the Faster-RCNN model, its NDOT varied depending on the number of +objects in the image. The graph neural network, unlike other conventional CNN, +can address this inconsistent number of inputs due to the flexible graph struc- +ture with nodes and edges. Unlike ALBEF, the dimensions of image features and +global features from LightningDOT were the same with d(DOT ) +v += d(DOT ) +h += 768. +In terms of text encoder, the output of both models was similar since they used +the same BERT network: d(ALB) +w += d(DOT ) +w += 768. We projected these features +to a latent space where dp = ds = 512, which were the average of their original +dimensions. We used a 1-layer GATv2 network with H = 4 multi-head atten- +tions to update the graph features while still keeping the input dimensions of +d′ +p = d′ +s = 512. We also applied Dropout with p = 0.7 in linear layers and +graph neural networks. In total, our HADA contained roughly 10M trainable +parameters. +The input pretrained models were pretrained on several large external datasets. +For example, ALBEF was pretrained on 14M images compared to only 29K im- +ages on Flickr30k that we used to train HADA. We used this advantage in our +prediction instead of train HADA in millions of samples. We modified the simi- +larity score to a weighted sum of our predictions and the original prediction of +the input models. Therefore, the weighted similarity score that we used was: +S(I , T) = (1 − α)⟨hp, hs⟩ + α⟨h(ALB) +v +, h(ALB) +w +⟩ +where α was a trainable parameter. We did not include the original result of the +LightningDOT model, since its result was lower than ALBEF by a large margin +and therefore could have a negative impact on overall performance8. +We trained HADA for 50 epochs (early stopping9 was implemented) using +the batch size of 20 on 1 NVIDIA RTX3080Ti GPU. We used the AdamW +[23] optimizer with a weight decay of 0.02. The learning rate was set at 1e−4 +and decayed to 5e−6 following cosine annealing [22]. Similarly to ALBEF, we +also applied RandAugment [4] for data augmentation. The initial temperature +parameter was 0.07 [36] and we kept it in range of [0.001, 0.5] during training. To +mitigate the dominant effect of ALBEF global features on our weighted similarity +score, we first trained HADA with α = 0. After the model had converged, we +continued to train, but initially set α = 0.5 and kept it in the range of [0.1, 0.9]. +8 We tried including the LightningDOT in the weighted similarity score, but the result +was lower than using only ALBEF. +9 In our experiment, it converged after roughly 20 epochs. + +HADA in Image-text Retrieval +11 +4.3 +Baselines +We built 2 baselines that also integrated ALBEF and LightningDOT as an input +to show the advantages of using graph structures to fuse these input models. +Baseline B1. We calculated the average of the original ranking results obtained +from ALBEF and LightningDOT and considered them as the distance between +images and text. This meant that the relevant pairs should be ranked at the top, +whilst irrelevant pairs would have lower places. +Baseline B2. Instead of using a graph structure to fuse the features extracted +from the pretrained models, we only concatenated their global embedding and +fed them into the last linear layers to obtain the unified features. We trained +this baseline B2 following the same strategy as described in Section 4.2 using +the weighted similarity score. +4.4 +Comparison to Baseline +Table 1 illustrated the evaluation metrics of the difference models in the Flickr30k +dataset. Similarly to LightningDOT, our main target was to introduce an image- +text retrieval model that did not implement a cross-modality transformer mod- +ule to ensure that it can perform in real-time without any delay. Thus, we only +reported the result from LightningDOT and ALBEF that did not use the time- +consuming compartment to rerank in the subsequent step. If the model has a +better initial result, it can have a better reranked result by using the cross- +modality transformer later. We also added UNITER [3] and VILLA [10], which +both used cross-modality transformer architecture to make the prediction, to the +comparison. +Table 1. Performance of models on Flickr30k Dataset. The symbol � indicated the +results were originally reported in their research, while others were from our re- +implementation using their public pretrained checkpoints. The column △R showed +the difference compared to ALBEF. +Methods +Image-to-Text +Text-to-Image +Total +△R +R@1 R@5 R@10 RSum R@1 +R@5 R@10 RSum +RSum +UNITER� +87.3 +98 +99.2 +284.5 75.56 94.08 96.76 +266.4 +550.9 +�13.68 +VILLA� +87.9 97.2 +98.8 +283.9 76.26 94.24 96.84 267.34 551.24 �13.34 +LightningDOT 83.6 +96 +98.2 +277.8 +69.2 +90.72 94.54 254.46 532.26 �32.32 +LightningDOT� 83.9 97.2 +98.6 +279.7 +69.9 +91.1 +95.2 +256.2 +535.9 +�28.68 +ALBEF +92.6 99.3 +99.9 +291.8 79.76 +95.3 +97.72 272.78 564.58 +0 +B1 +90.7 +99 +99.6 +289.3 79.08 +94.5 +96.94 270.52 559.82 +�4.76 +B2 +91.4 99.5 +99.7 +290.6 79.64 95.34 97.46 272.44 563.04 +�1.54 +HADA +93.3 99.6 100 292.9 81.36 95.94 98.02 275.32 568.22 �3.64 + +12 +Manh-Duy et al. +It was clearly that our HADA obtained the highest metrics at all recall values +compared to others. HADA achieved a slightly better R@5 and R@10 in Image- +to-Text (I2T) and Text-to-Image (T2I) subtasks than ALBEF. However, the +gap became more significant at R@1. We improved the R@1 of I2T by 0.7% +(92.96 → 93.3) and the R@1 of T2I by 1.6% (79.76 → 81.36). In total, our +RSum was 3.64% higher than that of ALBEF (564.58 → 568.22). +The experiment also showed that LightningDOT, which encoded images us- +ing Faster-RCNN, was much behind ALBEF when its total RSum was lower +than that of ALBEF by approximately 30%. The reason might be that the ob- +ject detector was not as powerful as the ViT network and LightningDOT was +pretrained on 4M images compared to 14M images used to train ALBEF. Al- +though also using object detectors as the backbone but applying a cross-modality +network, UNITER and VILLA surpassed LightningDOT by a large margin at +15%. It proved that this intensive architecture made the large impact on the +multimodal retrieval. +Regarding our 2 baselines B1 and B2, both of them were failed to get better +results than the input model ALBEF. Model B1, with the simple strategy of tak- +ing the average ranking results and having no learnable parameters, performed +worse than model B2 which used a trainable linear layer to fuse the pretrained +features. Nevertheless, the RSum of B2 was lower than HADA by 5.18%. It +showed the advantages of using graph structure to fuse the information between +models to obtain the better result. +4.5 +Ablation Study +To show the stable performance of HADA, we used it to combine 2 other different +pretrained models, including BLIP [17] and CLIP [28]. While CLIP is well-known +for its application in many retrieval challenges [24,32,9,31], BLIP is the enhanced +version of ALBEF with the bootstrapping technique in the training process. We +used the same configuration as described in 4.2 to train and evaluate HADA in +Flickr30k and MSCOCO datasets. We used the pretrained BLIP and CLIP from +LAVIS library [16]. It was noted that the CLIP we used in this experiment was +the zero-shot model, since the fine-tuned CLIP for these datasets is not available +yet. +Table 2 showed the comparison between HADA and the input models. CLIP +performed worst on both Flickr30k and MSCOCO with huge differences com- +pared to BLIP and HADA because CLIP was not fine-tuned for these datasets. +Regarding Flickr30k dataset, HADA managed to improve the RSum by more +than 3.9% compared to that of BLIP. Additionally, HADA obtained the highest +scores in all metrics for both subtasks. Our proposed framework also increased +the RSum of BLIP by 1.49% in MSCOCO dataset. However, BLIP performed +slightly better HADA in the I2T subtask while HADA achieved higher perfor- +mance in the T2I subtask. + +HADA in Image-text Retrieval +13 +Table 2. Performance of models on the test set in Flickr30k and MSCOCO datasets. +The column △R showed the difference compared to BLIP in that dataset. +Dataset Methods +Image-to-Text +Text-to-Image +Total +△R +R@1 R@5 R@10 RSum +R@1 +R@5 R@10 RSum +RSum +Flickr30k +BLIP +94.3 +99.5 +99.9 +293.7 +83.54 96.66 98.32 278.52 572.22 +0 +CLIP +88 +98.7 +99.4 +286.1 +68.7 +90.6 +95.2 +254.5 +540.6 +�31.62 +HADA +95.2 99.7 +100 +294.9 +85.3 97.24 98.72 281.26 576.16 �3.94 +MSCOCO +BLIP +75.76 93.8 96.62 266.18 57.32 81.84 88.92 228.08 494.26 +0 +CLIP +57.84 81.22 87.78 226.84 37.02 61.66 +71.5 +170.18 397.02 �97.24 +HADA +75.36 92.98 96.44 264.78 58.46 82.85 89.66 230.97 495.75 �1.49 +5 +Conclusion +In this research, we proposed a simple graph-based framework, called HADA, +to combine 2 pretrained models to address the image-text retrieval problem. We +created a graph structure to fuse the extracted features obtained from the pre- +trained models, followed by the GATv2 network to update them. Our proposed +HADA only contained roughly 10M learnable parameters, helping it become +easy to train using only 1 GPUs. Our experiments showed the promisingness of +the proposed method. Compared to input models, we managed to increase total +recall by more than 3.6%. Additionally, we implemented other 2 simple base- +lines to show the advantage of using the graph structures. This result helped +us resolve 2 questions: (1) increase the performance of SOTA models in image- +text retrieval task and (2) not requiring many GPUs to train on any large-scale +external dataset. It has opened the possibility of applying HADA in industry +where many small and medium start-ups do not possess many GPUs. +Although we achieved the better result compared to the baselines, there are +still rooms to improve the performance of HADA. Firstly, it can be extended not +only by 2 pretrained models as proposed in this research, but can be used with +more than that number. Secondly, the use of different graph neural networks, +such as the graph transformer [30], can be investigated in future work. Third, the +edge feature in the graph is also considered. Currently, HADA did not implement +the edge feature in our experiment, but they can be learnable parameters in +graph neural networks. Last but not least, pretraining HADA on a large-scale +external dataset as other SOTA have done might enhance its performance. +6 +Acknowledgement +This publication has emanated from research supported in party by research +grants from Science Foundation Ireland under grant numbers SFI/12/RC/2289, +SFI/13/RC/2106, and 18/CRT/6223. + +14 +Manh-Duy et al. +References +1. Brody, S., Alon, U., Yahav, E.: How attentive are graph attention networks? arXiv +preprint arXiv:2105.14491 (2021) +2. Chen, C.F.R., Fan, Q., Panda, R.: Crossvit: Cross-attention multi-scale vision +transformer for image classification. 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Young, P., Lai, A., Hodosh, M., Hockenmaier, J.: From image descriptions to visual +denotations: New similarity metrics for semantic inference over event descriptions. +Transactions of the Association for Computational Linguistics 2, 67–78 (2014) + diff --git a/WtE3T4oBgHgl3EQf1AtK/content/tmp_files/load_file.txt b/WtE3T4oBgHgl3EQf1AtK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3409ea8f873fa73230913d65a58024a5621ccd5e --- /dev/null +++ b/WtE3T4oBgHgl3EQf1AtK/content/tmp_files/load_file.txt @@ -0,0 +1,825 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf,len=824 +page_content='HADA: A Graph-based Amalgamation Framework in Image-text Retrieval Manh-Duy Nguyen1[0000−0001−6878−7039], Binh T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Nguyen2,3,4, and Cathal Gurrin1[0000−0003−2903−3968] 1 School of Computing, Dublin City University, Ireland 2 VNU-HCM, University of Science, Vietnam 3 Vietnam National University Ho Chi Minh City, Vietnam 4 AISIA Lab, Vietnam Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Many models have been proposed for vision and language tasks, especially the image-text retrieval task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' All state-of-the-art (SOTA) models in this challenge contained hundreds of millions of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' They also were pretrained on a large external dataset that has been proven to make a big improvement in overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' It is not easy to propose a new model with a novel architecture and intensively train it on a massive dataset with many GPUs to surpass many SOTA mod- els, which are already available to use on the Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' In this paper, we proposed a compact graph-based framework, named HADA, which can combine pretrained models to produce a better result, rather than build- ing from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' First, we created a graph structure in which the nodes were the features extracted from the pretrained models and the edges connecting them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' The graph structure was employed to capture and fuse the information from every pretrained model with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Then a graph neural network was applied to update the connection between the nodes to get the representative embedding vector for an image and text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Finally, we used the cosine similarity to match images with their rele- vant texts and vice versa to ensure a low inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Our experiments showed that, although HADA contained a tiny number of trainable pa- rameters, it could increase baseline performance by more than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='6% in terms of evaluation metrics in the Flickr30k dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Additionally, the proposed model did not train on any external dataset and did not require many GPUs but only 1 to train due to its small number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' The source code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='com/m2man/HADA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Keywords: image-text retrieval · graph neural network · fusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' 1 Introduction Image-text retrieval is one of the most popular challenges in vision and lan- guage tasks, with many state-of-the-art models (SOTA) recently introduced [19,3,18,28,10,25,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' This challenge includes 2 subtasks, which are image-to- text retrieval and text-to-image retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' The former subtask is defined as an arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='04742v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='CV] 11 Jan 2023 2 Manh-Duy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' image query is given to retrieve relevant texts in a multimodal dataset, while the latter is vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Most of the SOTA models in this research field shared 2 things in common: (1) they were built on transformer-based cross-modality attention architectures [3,19] and (2) they were pretrained on the large-scale multimodal data crawled from the Internet [28,18,19,17,13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' However, these things have their own disad- vantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' The attention structure between 2 modalities could achieve an accurate result, but it cost a large amount of inference time due to the massive compu- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' For instance, UNITER [3] contained roughly 303 millions parameters and it took a decent amount of time to perform the retrieval in real-time [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Many recent work has resolved this model-related problem by introducing joint- encoding learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' They can learn visual and semantic information from both modalities without using any cross-attention modules, which can be applied later to rerank the initial result [18,25,31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Figure 1 illustrated the architecture of these pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Regarding the data perspective, the large collected data usu- ally come with noisy annotation, and hence could be harmful to the models that are trained on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Several techniques have been proposed to mitigate this issue [19,18,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' However, training on the massive dataset still creates a burden on the computation facility, such as the number of GPUs, which are required to train the model successfully and efficiently [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Two most popular pipelines of the SOTA for image-text retrieval challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' (a) A cross-modality transformer network is applied to measure the similarity between an image and a text based on their features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' (b) Each modality used their own transformer network to get its global embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' It has motivated us to answer the question: Can we combine many SOTA models, which are currently available to use, to get a better unified model without intensively training with many GPUs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' In this paper, we introduced a graph- based amalgamation framework, called HADA, which formed a graph-based structure to fuse the features produced by other pretrained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' We did not Similarity Score Similarity Score Linear Layers Linear Linear Layers Layers Cross-modality Transformer ImageTransformer Text Transformer ImageFeature Global ImageFeature Image Encoder Text Encoder Image Encoder Text Encoder TextFeature GlobalTextFeature Image Input Text Input [CLS] Feature Image Input Text Input (a) (b)HADA in Image-text Retrieval 3 use any time-consuming cross-modality attention network to ensure fast retrieval speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' A graph neural network was employed to extract visual and textual em- bedded vectors from fused graph-based structures of images and texts, where we can measure their cosine similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' To the best of our knowledge, the graph struc- ture has been widely applied in the image-text retrieval challenge [26,7,27,35,21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Nevertheless, it was utilized to capture the interaction between objects or align local and global information within images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' HADA is the first approach that applies this data structure to combine SOTA pretrained models by fusing their features in each modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' We trained HADA only on the Flickr30k dataset with- out using any large-scale datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' We applied Momentum Distillation technique [18], which was shown that can not only mitigate the harmful effect of noise an- notation but also improve the accuracy on a clean dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Our experiments showed that HADA, with the tiny extra number of training parameters, could improve total recalls by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='64% compared to the input SOTA without training with millions of additional image-text pairs as other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' This is the most crucial part since it is not easy to possess multiple GPUs to use, especially for small and medium businesses or start-up companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Therefore, we believe that HADA can be applied not only in the academic field, but also in industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Our main contribution can be summarised as follow: (1) We introduced HADA, a compact pipeline that can combine 2 or many SOTA pretrained mod- els to address the image-text retrieval challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' (2) We proposed a way to fuse the information between input pretrained models by using graph structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' (3) We evaluated the performance of HADA on the well-known Flickr30k dataset [37] and MSCOCO dataset [20] without using any other large-scale dataset but still improved the accuracy compared to the baseline input models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' 2 Related Work A typical vision-and-language model, including image-text retrieval task, was built with the usage of transformer-based encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' In specific, OSCAR [19], UNITER [3], and VILLA [10] firstly employed Faster-RCNN [29] and BERT [6] to extract visual and text features from images and texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' These features were then fed into a cross-modality transformer block to learn the contextualized embedding that captured the relations between regional features from images and word pieces from texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' An additional fully connected layer was used to classify whether the images and texts were relevant to each other or not, based on the embedding vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Although achieving superior results, these approaches had a drawback of being applied to real-time use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' It required a huge amount of time to perform the retrieval online, since the models have to process the intensive cross-attention transformer architecture many times for every single query [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Recently, there have been some works proposing an approach to resolve that problem by utilising 2 distinct encoders for images and text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' The data from each modality now can be embedded offline and hence improve the re- trieval speed [31,18,17,25,13,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' In terms of architecture, all approaches used 4 Manh-Duy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' the similar BERT-based encoder for semantic data but different image encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' While LightningDOT [31] encoded images with detected objects extracted by the Faster-RCNN model, FastnSlow [25] applied the conventional Resnet net- work to embed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' On the other side, ALBEF [18] and BLIP [17] employed the Vision Transformer backbone [8] to get the visual features corresponding to their patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Because these SOTA did not use the cross-attention struc- ture, which was a critical point to achieve high accuracy, they applied different strategies to increase performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Specifically, pretraining a model on a large dataset can significantly improve the result [18,19,13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' For instance, CLIP [28] and ALIGN [13] were pretrained on 400 millions and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='8 billions image-text pairs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Another way was that they ran another cross-modality image-text retrieval model to rerank the initial output and get a more accurate result [18,31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Regarding to graph structures, SGM [35] introduced a visual graph encoder and a textual graph encoder to capture the interaction between objects appear- ing in images and between the entities in text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' LGSGM [26] proposed a graph embedding network on top of SGM to learn both local and global information about the graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Similarly, GSMN [21] presented a novel technique to assess the correspondence of nodes and edges of graphs extracted from images and texts separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' SGRAF [7] build a reasoning and filtration graph network to refine and remove irrelevant interactions between objects in both modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Although there are many SOTAs with different approaches for image-text retrieval problems, there is no work that tries combining these models but intro- ducing a new architecture and pretrain on a massive dataset instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Training an entire new model from scratch on the dataset is not an easy challenge since it will create a burden on the computation facilities such as GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' In this paper, we introduced a simple method which combined the features extracted from the pretrained SOTA by applying graph structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Unlike other methods that also used this data structure, we employed graphs to fuse the information between the input features, which was then fed into a conventional graph neural network to obtain the embedding for each modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Our HADA consisted of a small number of trainable parameters, hence can be easily trained on a small dataset but still obtained higher results compared to the input models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' 3 Methodology This section will describe how our HADA addressed the retrieval challenge by combining any available pretrained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Figure 2 depicted the workflow of HADA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' We started with only 2 models (Nmodels = 2) as illustrated in Figure 2 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Nevertheless, HADA can be extended with a larger Nmodels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' HADA began with using some pretrained models to extract the features from each modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' We then built a graph structure to connect the extracted fea- tures together, which were fed into a graph neural network (GNN) later to update them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' The outputs of the GNN were concatenated with the original global features produced by the pretrained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Finally, simple linear layers were employed at the end to get the final representation embedding features for HADA in Image-text Retrieval 5 images and texts, which can be used to measure the similarity to perform the retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' For evaluation, we could extract our representation features offline to guarantee the high speed inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' The pipeline of the proposed HADA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' The red borders indicated trainable com- ponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' The ITM and ITC infered the training tasks which will be discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='1 Revisit State-of-the-art Models We only used the pretrained models without using the cross-modality trans- former structure to extract the features as depicted in Figure 1 to reduce the number of computations and ensure the high speed inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Basically, they used a unimodal encoder to get the features of an image or a text followed by a transformer network to embed them and obtain the [CLS] embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' This [CLS] token was updated by 1 or many fully connected layers to become a representative global feature that can be compared with that of the remaining modality to get the similarity score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' HADA began with the output of the transformer layer from the pretrained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' In detail, for an input image I, we obtained the sequence of patch tokens from each model i denoted as v(i) = {v(i) cls, v(i) 1 , v(i) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=', v(i) Ni}, where v(i) j ∈ Rd(i) v and Ni was the length of the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' This length depended on the architecture of the image encoder network employed in the pretrained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' For example, it could be the number of patches if the image encoder was a Vision Transformer (ViT) network [8], or the number of detected objects or regions of interest if the encoder was a Faster-RCNN model [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Additionally, we also extracted the global visual representation feature h(i) v ∈ Rd(i) h from v(i) cls as illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Regarding the semantic modality, we used the same process as that of the visual modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Specifically, we extracted the sequence of patch tokens w(i) = {w(i) cls, w(i) 1 , w(i) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' w(i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='L } where w(i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='∈ Rd(i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='and L was the length of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Pretrained ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Model 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Creating ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='LinearLayers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Graph Structure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Graph Neural ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Linear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Image Input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Layers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Linear Layers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Pretrained ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Model 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='ITC / ITM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Pretrained ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Model 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Creating ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='LinearLayers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Graph Structure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Graph Neural ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Linear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Text Input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Layers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Linear Layers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Pretrained ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Model 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Image Feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='[CLS] Image Feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Global Image Feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Final Image Feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Text Feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='[CLS] Text Feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Global Text Feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Final Text Feature6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='Manh-Duy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' the text, and the global textual representation embedding h(i) w ∈ Rd(i) h for an input text T using the pretrained model i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' The input model i matched a pair of an image I and a text T by calculating the dot product ⟨h(i) v , h(i) w ⟩ of their global features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' However, HADA not only used the global embedding but also the intermediate transformer tokens to make the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' We used our learned [CLS] tokens to improve the global features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' In contrast, using the original global features could ensure high performance of the pretrained models and mitigate the effect of unhelpful tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='2 Create Graph Structure Each pretrained model i produced different [CLS] features v(i) cls and w(i) cls for an image and text, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Since our purpose was to combine the models, we needed to fuse these [CLS] tokens to obtain the unified ones for each modality separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' In each modality, for example, the visual modality, HADA not only updated v(i) cls based on v(i) solely but also on those of the remaining pretrained models {v(j) | j ̸= i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Because these v came from different models, their dimen- sions could be not similar to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Therefore, we applied a list of linear layers f (i) v : Rd(i) v → Rdp to map them in the same dimensional space: p(i) = {f (i) v (x)|x ∈ v(i)} = {p(i) cls, p(i) 1 , p(i) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=', p(i) Ni} We performed a similar process for the textual modality to obtain: s(i) = {f (i) w (x)|x ∈ w(i)} = {s(i) cls, s(i) 1 , s(i) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=', s(i) L }, where f (i) w : Rd(i) w → Rds We then used graph structures Gp = {Vp, Ep} and Gs = {Vs, Es} to connect these mapped features together, where V and E denoted the list of nodes and edges in the graph G accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' In our HADA, nodes indicated the mapped features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Specifically, Vp = {p(i)} and Vs = {s(i)} for all i ∈ [1, Nmodels].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Re- garding edges,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' we symbolized ea→b as a directed edge from node a to node b in the graph,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' thus the set of edges of the visual graph Ep and the textual graph Es were: Ep = {ex→p(j) cls | x ∈ p(i) and i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' j ∈ [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Nmodels]} Es = {ex→s(j) cls | x ∈ s(i) and i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' j ∈ [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Nmodels]} To be more detailed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' we created directed edges that went from every patch features to the [CLS] feature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' including from the [CLS] itself,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' for all pretrained models but not in the reversed direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' The reason was that [CLS] was originally introduced as a representation of all input data, so it would summarize all patch tokens [8,2,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Therefore, it would be the node that received information from other nodes in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' This connection structure ensured that HADA can update the [CLS] tokens based on the patch tokens from all pretrained models in a fine-grained manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' HADA in Image-text Retrieval 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='3 Graph Neural Network Graph neural networks (GNN) have witnessed an increase in its popularity over the past few years, with many GNN structures having been introduced re- cently [15,5,34,11,30,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' HADA applied the modified Graph Attention Network (GATv2), which was recommended to be used as a baseline whenever employing GNN [1], to fuse the patch features from different pretrained models together to get the unified [CLS] features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Let Nk = {x ∈ V | ex→k ∈ E} be the set of neighbor nodes from which there was an edge connecting to node k in the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' GATv2 used a scoring function se to weight every edge indicating the importance of the neighbor nodes x in Nk before updating the node k ∈ Rd: se(ex→k) = A⊤LeakyRELU(W1x + W2k]) where A ∈ Rd′, W1 ∈ Rd′×d, and W2 ∈ Rd′×d were learnable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' These weights were then normalized across all neighbor nodes in Nk by using a softmax function to get the attention scores: αex→k = exp(se(ex→k)) � y∈Nk exp(se(ey→k)) The updated node k′ ∈ Rd′ was then calculated based on its neighbors in Nk, including k if we add an edge connect it to itself: k′ = σ( � x∈Nk αex→k · W1x) where σ was an nonlinearity activate function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Furthermore, this GATv2 network could be enlarged by applying a multi-head attention structure and improved performance [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' The output now was a concatenation of each head output, which was similar to Transformer architecture [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' An extra linear layer was used at the end to convert these concatenated nodes to the desired dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' We used distinct GATv2 structures with H attention heads for each modality in this stage, as illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' HADA took the input graphs Gp and Gs with nodes Vp and Vs in the vector space of dp and ds dimensions and updated them to V′p = {p′(i)} and V′s = {s′(i)} with dimensions of d′ p and d′ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' We then concatenated the updated [CLS] nodes p′ cls and s′ cls from all pretrained models with their corresponding original global embedding hv and hw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Finally, we fed them into a list of linear layers to get our normalized global representation hp ∈ Rdh and hs ∈ Rdh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='4 Training Tasks Image-Text Contrastive Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' HADA encoded the input image I and text T to hp and hs, accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' We used a similarity function that was a dot product S(I , T) = ⟨hp, hs⟩ = h⊤ p hs to ensure that a pair of relevant image-text (positive pair) would have a higher similar representation compared to irrelevant 8 Manh-Duy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' pairs (negative pairs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' The contrastive loss for image-to-text (i2t) retrieval and text-to-image (t2i) retrieval for the mini-batch of M relevant pairs (I m, T m) were: Li2t(I m) = −log exp(S(I m, T m)/τ) �M i=1 exp(S(I m, T i)/τ) Lt2i(T m) = −log exp(S(T m, I m)/τ) �M i=1 exp(S(T m, I i)/τ) where τ was a temperature parameter that could be learned during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' This contrastive learning has been used in many vision-and-language models and has been proven to be effective [18,31,17,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' In our experiment, we trained HADA with the loss that optimized both subtasks: LIT C = 1 M M � m=1 (Li2t(I m) + Lt2i(T m)) Inspired by ALBEF [18], we also applied momentum contrast (MoCo) [12] and their momentum distillation strategy for this unsupervised representation learning to cope with the problem of noisy information in the dataset and im- prove accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Image-Text Matching This objective was a binary classification task to dis- tinguish irrelevant image-text pairs, but were similar representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' This task would ensure that they were different in fine-grained details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' We implemented an additional disciminator layer dc : R4dh → R on top of the final embedding features hp and hs to classify whether the image I and the text T is a positive pair or not: dc(hp, hs) = sigmoid(W⊤[hp∥hs∥abs(hp − hs)∥hp ⊙ hs]) where W ∈ R4dh was trainable parameters, ∥ indicated the concatenation, abs(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=') was the absolute value, and ⊙ denoted elementwise multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' We used binary cross-entropy loss for this ordinary classification task: Litm(I , T) = ylog(dc(hp, ds)) + (1 − y)log(1 − dc(hp, ds)) where y was the one-hot vector representing the ground truth label of the pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' For each positive pair in the minibatch of M positive pairs, we sampled 1 hard negative text for the image and 1 hard negative image for the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' These negative samples were chosen from the current mini-batch in which they were not relevant based on the ground-truth labels, but have the highest similarity dot product score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Therefore, the objective for this task was: LIT M = 1 3M M � m=1 (Litm(I m, T m) + Litm(I m, T ′ m) + Litm(I ′ m, T m)) HADA in Image-text Retrieval 9 where T ′ m and I ′ m were the hard negative text and image samples in the mini- batch that were corresponding with the I m and T m, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' The final loss function in HADA was: L = LIT C + LIT M 4 Experiment 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='1 Dataset and Evaluation Metrics We trained and evaluated HADA on 2 different common datasets in the image- text retrieval task which are Flickr30k [37] and MSCOCO [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Flickr30k dataset consists of 31K images collected on the Flickr website, while MSCOCO comprises 123K images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Each image contains 5 relevant texts or captions that describe the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' We used Karpathy’s split [14], which has been widely applied by all mod- els in the image-text retrieval task, to split each dataset into train/evaluate/test on 29K/1K/1K and 113K/5K/5K images on Flickr30k and MSCOCO, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' The common evaluation metric in this task is the Recall at K (R@K) when many SOTAs used this metric [18,31,17,28,13,19,3,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' This metric is defined as the proportion of the number of queries that we found the correct relevant output in the top K of the retrieved ranked list: R@K = 1 Nq Nq � q=1 1(q, K) where Nq is the number of queries and 1(q, K) is a binary function returning 1 if the model find the correct answer of the query q in the top K of the retrieved output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' In particular, for the image-to-text subtask, R@K is the percentage of the number of images where we found relevant texts in the top K of the output result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' In our experiment, we used R@1, R@5, R@10, and RSum which was the sum of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='2 Implementation Details In our experiment, we combined 2 SOTA models that had available pretrained weights fine-tuned on the Flickr30k dataset: ALBEF5 and LightningDOT6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' None of them used the cross-modality transformer structure when retrieved to ensure the fast inference speed7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Although they used the same BERT architecture to encode a text, the former model employed the ViT network to encode an im- age, while the latter model applied the Faster-RCNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' We chose these 2 5 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='com/salesforce/ALBEF 6 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='com/intersun/LightningDOT 7 Indeed, these 2 models applied the cross-modality transformer network to rerank the initial result in the subsequent step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' However, we did not focus on this stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' 10 Manh-Duy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' models because we wanted to combine different models with distinct embedding backbones to utilize the advantages of each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Regarding ALBEF, their ViT network encoded an image to 577 patch to- kens including the [CLS] one (NALB = 576 and d(ALB) v = 768).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' This [CLS] was projected to the lower dimension to obtain the global feature (d(ALB) h = 256).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Because LightningDOT encoded an image based on the detected objects pro- duced by the Faster-RCNN model, its NDOT varied depending on the number of objects in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' The graph neural network, unlike other conventional CNN, can address this inconsistent number of inputs due to the flexible graph struc- ture with nodes and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Unlike ALBEF, the dimensions of image features and global features from LightningDOT were the same with d(DOT ) v = d(DOT ) h = 768.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' In terms of text encoder, the output of both models was similar since they used the same BERT network: d(ALB) w = d(DOT ) w = 768.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' We projected these features to a latent space where dp = ds = 512, which were the average of their original dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' We used a 1-layer GATv2 network with H = 4 multi-head atten- tions to update the graph features while still keeping the input dimensions of d′ p = d′ s = 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' We also applied Dropout with p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='7 in linear layers and graph neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' In total, our HADA contained roughly 10M trainable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' The input pretrained models were pretrained on several large external datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' For example, ALBEF was pretrained on 14M images compared to only 29K im- ages on Flickr30k that we used to train HADA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' We used this advantage in our prediction instead of train HADA in millions of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' We modified the simi- larity score to a weighted sum of our predictions and the original prediction of the input models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Therefore, the weighted similarity score that we used was: S(I , T) = (1 − α)⟨hp, hs⟩ + α⟨h(ALB) v , h(ALB) w ⟩ where α was a trainable parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' We did not include the original result of the LightningDOT model, since its result was lower than ALBEF by a large margin and therefore could have a negative impact on overall performance8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' We trained HADA for 50 epochs (early stopping9 was implemented) using the batch size of 20 on 1 NVIDIA RTX3080Ti GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' We used the AdamW [23] optimizer with a weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' The learning rate was set at 1e−4 and decayed to 5e−6 following cosine annealing [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Similarly to ALBEF, we also applied RandAugment [4] for data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' The initial temperature parameter was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='07 [36] and we kept it in range of [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='5] during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' To mitigate the dominant effect of ALBEF global features on our weighted similarity score, we first trained HADA with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' After the model had converged, we continued to train, but initially set α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='5 and kept it in the range of [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' 8 We tried including the LightningDOT in the weighted similarity score, but the result was lower than using only ALBEF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' 9 In our experiment, it converged after roughly 20 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' HADA in Image-text Retrieval 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='3 Baselines We built 2 baselines that also integrated ALBEF and LightningDOT as an input to show the advantages of using graph structures to fuse these input models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Baseline B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' We calculated the average of the original ranking results obtained from ALBEF and LightningDOT and considered them as the distance between images and text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' This meant that the relevant pairs should be ranked at the top, whilst irrelevant pairs would have lower places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Baseline B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Instead of using a graph structure to fuse the features extracted from the pretrained models, we only concatenated their global embedding and fed them into the last linear layers to obtain the unified features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' We trained this baseline B2 following the same strategy as described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='2 using the weighted similarity score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='4 Comparison to Baseline Table 1 illustrated the evaluation metrics of the difference models in the Flickr30k dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Similarly to LightningDOT, our main target was to introduce an image- text retrieval model that did not implement a cross-modality transformer mod- ule to ensure that it can perform in real-time without any delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Thus, we only reported the result from LightningDOT and ALBEF that did not use the time- consuming compartment to rerank in the subsequent step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' If the model has a better initial result, it can have a better reranked result by using the cross- modality transformer later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' We also added UNITER [3] and VILLA [10], which both used cross-modality transformer architecture to make the prediction, to the comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Performance of models on Flickr30k Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' The symbol � indicated the results were originally reported in their research, while others were from our re- implementation using their public pretrained checkpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' The column △R showed the difference compared to ALBEF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Methods Image-to-Text Text-to-Image Total △R R@1 R@5 R@10 RSum R@1 R@5 R@10 RSum RSum UNITER� 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='3 98 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='2 284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='5 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='56 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='08 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='76 266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='4 550.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='9 �13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='68 VILLA� 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='9 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='2 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='8 283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='9 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='26 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='24 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='84 267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='34 551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='24 �13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='34 LightningDOT 83.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='26 �32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='32 LightningDOT� 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='9 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='2 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='6 279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='7 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='9 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='1 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='2 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='2 535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='9 �28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='68 ALBEF 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='6 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='3 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='9 291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='8 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='76 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='3 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='72 272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='78 564.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='58 0 B1 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='7 99 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='6 289.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='3 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='08 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='5 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='94 270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='52 559.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='82 �4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='76 B2 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='4 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='5 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='7 290.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='6 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='64 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='34 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='46 272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='44 563.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='04 �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='54 HADA 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='3 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='6 100 292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='9 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='36 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='94 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='02 275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='32 568.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='22 �3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='64 12 Manh-Duy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' It was clearly that our HADA obtained the highest metrics at all recall values compared to others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' HADA achieved a slightly better R@5 and R@10 in Image- to-Text (I2T) and Text-to-Image (T2I) subtasks than ALBEF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' However, the gap became more significant at R@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' We improved the R@1 of I2T by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='7% (92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='96 → 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='3) and the R@1 of T2I by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='6% (79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='76 → 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' In total, our RSum was 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='64% higher than that of ALBEF (564.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='58 → 568.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' The experiment also showed that LightningDOT, which encoded images us- ing Faster-RCNN, was much behind ALBEF when its total RSum was lower than that of ALBEF by approximately 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' The reason might be that the ob- ject detector was not as powerful as the ViT network and LightningDOT was pretrained on 4M images compared to 14M images used to train ALBEF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Al- though also using object detectors as the backbone but applying a cross-modality network, UNITER and VILLA surpassed LightningDOT by a large margin at 15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' It proved that this intensive architecture made the large impact on the multimodal retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Regarding our 2 baselines B1 and B2, both of them were failed to get better results than the input model ALBEF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Model B1, with the simple strategy of tak- ing the average ranking results and having no learnable parameters, performed worse than model B2 which used a trainable linear layer to fuse the pretrained features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Nevertheless, the RSum of B2 was lower than HADA by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='18%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' It showed the advantages of using graph structure to fuse the information between models to obtain the better result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='5 Ablation Study To show the stable performance of HADA, we used it to combine 2 other different pretrained models, including BLIP [17] and CLIP [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' While CLIP is well-known for its application in many retrieval challenges [24,32,9,31], BLIP is the enhanced version of ALBEF with the bootstrapping technique in the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' We used the same configuration as described in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='2 to train and evaluate HADA in Flickr30k and MSCOCO datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' We used the pretrained BLIP and CLIP from LAVIS library [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' It was noted that the CLIP we used in this experiment was the zero-shot model, since the fine-tuned CLIP for these datasets is not available yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Table 2 showed the comparison between HADA and the input models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' CLIP performed worst on both Flickr30k and MSCOCO with huge differences com- pared to BLIP and HADA because CLIP was not fine-tuned for these datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Regarding Flickr30k dataset, HADA managed to improve the RSum by more than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='9% compared to that of BLIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Additionally, HADA obtained the highest scores in all metrics for both subtasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Our proposed framework also increased the RSum of BLIP by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='49% in MSCOCO dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' However, BLIP performed slightly better HADA in the I2T subtask while HADA achieved higher perfor- mance in the T2I subtask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' HADA in Image-text Retrieval 13 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Performance of models on the test set in Flickr30k and MSCOCO datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' The column △R showed the difference compared to BLIP in that dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Dataset Methods Image-to-Text Text-to-Image Total △R R@1 R@5 R@10 RSum R@1 R@5 R@10 RSum RSum Flickr30k BLIP 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='3 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='5 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='9 293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='7 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='54 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='66 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='32 278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='52 572.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='22 0 CLIP 88 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='7 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='4 286.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='1 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='7 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='6 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='2 254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='5 540.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='6 �31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='62 HADA 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='2 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='7 100 294.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='9 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='3 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='24 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='72 281.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='26 576.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='16 �3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='94 MSCOCO BLIP 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='76 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='8 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='62 266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='18 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='32 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='84 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='92 228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='08 494.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='26 0 CLIP 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='84 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='22 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='78 226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='84 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='02 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='66 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='5 170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='18 397.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='02 �97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='24 HADA 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='36 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='98 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='44 264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='78 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='46 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='85 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='66 230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='97 495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='75 �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='49 5 Conclusion In this research, we proposed a simple graph-based framework, called HADA, to combine 2 pretrained models to address the image-text retrieval problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' We created a graph structure to fuse the extracted features obtained from the pre- trained models, followed by the GATv2 network to update them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Our proposed HADA only contained roughly 10M learnable parameters, helping it become easy to train using only 1 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Our experiments showed the promisingness of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Compared to input models, we managed to increase total recall by more than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Additionally, we implemented other 2 simple base- lines to show the advantage of using the graph structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' This result helped us resolve 2 questions: (1) increase the performance of SOTA models in image- text retrieval task and (2) not requiring many GPUs to train on any large-scale external dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' It has opened the possibility of applying HADA in industry where many small and medium start-ups do not possess many GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Although we achieved the better result compared to the baselines, there are still rooms to improve the performance of HADA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Firstly, it can be extended not only by 2 pretrained models as proposed in this research, but can be used with more than that number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Secondly, the use of different graph neural networks, such as the graph transformer [30], can be investigated in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Third, the edge feature in the graph is also considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Currently, HADA did not implement the edge feature in our experiment, but they can be learnable parameters in graph neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' Last but not least, pretraining HADA on a large-scale external dataset as other SOTA have done might enhance its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' 6 Acknowledgement This publication has emanated from research supported in party by research grants from Science Foundation Ireland under grant numbers SFI/12/RC/2289, SFI/13/RC/2106, and 18/CRT/6223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} +page_content=' 14 Manh-Duy et al.' 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Linguistics 2, 67–78 (2014)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQf1AtK/content/2301.04742v1.pdf'} diff --git a/XdE3T4oBgHgl3EQf1Qur/content/tmp_files/2301.04745v1.pdf.txt b/XdE3T4oBgHgl3EQf1Qur/content/tmp_files/2301.04745v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e4ccaef9716f0aed6c1f51bacac4ee253de9bb69 --- /dev/null +++ b/XdE3T4oBgHgl3EQf1Qur/content/tmp_files/2301.04745v1.pdf.txt @@ -0,0 +1,135 @@ +Fast persistent homology computation +for functions on R +Marc Glisse∗ +January 13, 2023 +Abstract +0-dimensional persistent homology is known, from a computational +point of view, as the easy case. Indeed, given a list of n edges in non- +decreasing order of filtration value, one only needs a union-find data struc- +ture to keep track of the connected components and we get the persistence +diagram in time O(nα(n)). The running time is thus usually dominated +by sorting the edges in Θ(n log(n)). A little-known1 fact is that, in the +particularly simple case of studying the sublevel sets of a piecewise-linear +function on R or S1, persistence can actually be computed in linear time. +This note presents a simple algorithm that achieves this complexity. An +implementation will soon be available in Gudhi [1]. +1 +Main idea +The piecewise-linear (PL) function f : R → R is defined by its image at each +vertex, represented as an array A. Usual algorithms first replace this with a +lower-star filtration defined on a path graph (which is a special case of simplicial +complex and cubical complex). However, this is not needed for our approach +which works at the level of PL functions until the end. +We call a pattern several consecutive elements in an array whose values have +the same order as the name of the pattern. For instance, if A[i+1] < A[i+2] < +A[i], we have a pattern 312. +Note that if we have two identical consecutive values (pattern 11), it is fine +to keep only one, this does not affect the persistence diagram. Also, for two +(not necessarily consecutive) identical values, the stability theorem tells us that +it is fine to simulate simplicity by assuming for instance that the second one is +larger, so we do not need to consider patterns like 121 but only 132 or 231. The +last trivial remark is that in a pattern 123, we can drop the 2. In particular, we +only need to handle sequences of alternating local minima and local maxima. +The main ingredient is that when we see a pattern 1324 (or its symmetric +4231), the elements represented by 2 and 3 in the pattern define a pair in the +persistence diagram. Indeed, looking at the sublevel sets, a connected compo- +nent appears at 2, and at 3 it merges with an older component that has existed +∗marc.glisse@inria.fr. +Universit´e +Paris-Saclay, +CNRS, +Inria, +Laboratoire +de +Math´ematiques d’Orsay, 91405, Orsay, France. +1I would welcome a good reference, I wrote this note because I failed to locate one. +1 +arXiv:2301.04745v1 [cs.CG] 11 Jan 2023 + += ++ +1 +2 +3 +4 +Figure 1: Pattern 1324 and its reduction. +narrowing +expanding +Figure 2: A 2-phase sequence. +since at least 1. We can thus remove those two elements, 4 now directly follows +1, and the persistence diagram of the reduced array plus the pair (2, 3) is equal +to the persistence diagram of the original array, see Figure 1.. +After reducing the sequence based on those remarks, we are left with a +sequence with no 123, 321, 1324 or 4231 pattern. It is easy to see that such a +sequence has a very specific 2-phase shape depicted in Figure 2: it alternates +between local minima and local maxima, in a first expanding phase the maxima +are increasing and the minima decreasing, and in a second narrowing phase the +maxima are decreasing and the minima increasing (of course one of the phases +may be empty). Indeed, as soon as we see a pattern 132, the next element can +be neither lower than 2 (pattern 321) nor higher than 3 (pattern 1324), so the +triple that follows and shares 2 elements with 132 can only follow the pattern +312. Similarly, 312 can only be followed by 132, and the narrowing pattern +only stops at the end of the sequence. The expanding phase is symmetric to +the narrowing one, and we only need to notice that 2 consecutive local minima +must be in a pattern 132 or 231 to conclude. +The extremities also provide some opportunities. For instance if the sequence +starts with 21, we can remove 2. If it starts with 231, we can pair off 2 and 3, +remove them and start the sequence at 1. Symmetric operations are possible at +the other extremity. This is sufficient to reduce a 2-phased sequence as obtained +above to just one point, the global minimum, and we have the whole persistence +diagram. +To apply these operations efficiently, we propose adding the values one by +one from left to right, maintaining a reduced sequence with the invariant that +it has a narrowing pattern and starts by 12 (unless it is reduced to a single +2 + +value). This invariant is equivalent to the absence of patterns 123, 321, 1324 +and 4231, and of patterns 21 and 231 at the beginning. With every new value, +we check if appending it breaks the invariant, or equivalently if a simplification +involving the new element is possible, and we simplify until the invariant is +restored. After processing the whole input, and after removing the last element +in case the reduced sequence ends in 12, we can just pair and remove the last +2 elements (terminating 132 pattern) recursively until only 1 element remains. +Since we only go back when removing elements, the amortized complexity per +element is constant and the whole algorithm takes linear time. +2 +Function on a circle +For a PL function defined on the circle S1, we do not have extremities at which +we could simplify, but that is unnecessary, since removing the patterns 123/321 +and 1324/4231 is sufficient to get down to just 2 values. Indeed, without 123 +and 321, the sequence has an even length and alternates between local minima +and maxima. If the sequence has length at least 4, 2 consecutive minima are +involved in a pattern 132 or 231. By symmetry, we assume there is a pattern +132. As in the line case, when a narrowing pattern starts, it cannot end until the +end of the sequence. However, on a circle, the sequence is periodic and does not +end. In particular, it reaches this 1 again. In a narrowing sequence, the minima +increase, so when we reach 1 again, it must be larger than 2, a contradiction. +From an algorithmic point of view, we can cut the circle at an arbitrary point, +make a first pass to get a 2-phase pattern, then reconnect the 2 extremities and +simplify at the junction until we have only 2 values left. It may be convenient +to use the global minimum (or maximum, or both) as initial cutting point, +although it does not change the linear complexity. +3 +Parallelism +Although we expect this approach is fast enough in practice not to require +parallelization, it is tempting to try it. We can split the segment into smaller +segments, simplify each of them to a 2-phase shape in parallel, and collect the +pairs the simplifications find. And we can iterate, possibly using the point where +the phase changes (the minimum) inside each segment as the new splitting +points, or merging adjacent segments. For a fairly nice function, this should +reduce the sequence significantly and let us finish sequentially. +However, if +for instance the input already has a narrowing shape from the beginning, the +parallel phases will do almost nothing. This is then just a heuristic. There +is little hope of a perfectly parallel algorithm because of the non-locality of +persistent homology, Figure 3 shows that for a narrowing shape, adding at the +end a very large or very small value can change the pairing of all the points. +References +[1] The GUDHI Project. GUDHI User and Reference Manual. GUDHI Editorial +Board. URL: https://gudhi.inria.fr/doc/latest/. +3 + +Figure 3: Non-locality: the last element may determine the pairing of all the +other points. +4 + diff --git a/XdE3T4oBgHgl3EQf1Qur/content/tmp_files/load_file.txt b/XdE3T4oBgHgl3EQf1Qur/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f6f275786a64825e3bfb1bbcdf046aa4ee955612 --- /dev/null +++ b/XdE3T4oBgHgl3EQf1Qur/content/tmp_files/load_file.txt @@ -0,0 +1,68 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf,len=67 +page_content='Fast persistent homology computation for functions on R Marc Glisse∗ January 13, 2023 Abstract 0-dimensional persistent homology is known, from a computational point of view, as the easy case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' Indeed, given a list of n edges in non- decreasing order of filtration value, one only needs a union-find data struc- ture to keep track of the connected components and we get the persistence diagram in time O(nα(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' The running time is thus usually dominated by sorting the edges in Θ(n log(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' A little-known1 fact is that, in the particularly simple case of studying the sublevel sets of a piecewise-linear function on R or S1, persistence can actually be computed in linear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' This note presents a simple algorithm that achieves this complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' An implementation will soon be available in Gudhi [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' 1 Main idea The piecewise-linear (PL) function f : R → R is defined by its image at each vertex, represented as an array A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' Usual algorithms first replace this with a lower-star filtration defined on a path graph (which is a special case of simplicial complex and cubical complex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' However, this is not needed for our approach which works at the level of PL functions until the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' We call a pattern several consecutive elements in an array whose values have the same order as the name of the pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' For instance, if A[i+1] < A[i+2] < A[i], we have a pattern 312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' Note that if we have two identical consecutive values (pattern 11), it is fine to keep only one, this does not affect the persistence diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' Also, for two (not necessarily consecutive) identical values, the stability theorem tells us that it is fine to simulate simplicity by assuming for instance that the second one is larger, so we do not need to consider patterns like 121 but only 132 or 231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' The last trivial remark is that in a pattern 123, we can drop the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' In particular, we only need to handle sequences of alternating local minima and local maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' The main ingredient is that when we see a pattern 1324 (or its symmetric 4231), the elements represented by 2 and 3 in the pattern define a pair in the persistence diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' Indeed, looking at the sublevel sets, a connected compo- nent appears at 2, and at 3 it merges with an older component that has existed ∗marc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content='glisse@inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content='fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' Universit´e Paris-Saclay, CNRS, Inria, Laboratoire de Math´ematiques d’Orsay, 91405, Orsay, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' 1I would welcome a good reference, I wrote this note because I failed to locate one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content='04745v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content='CG] 11 Jan 2023 = + 1 2 3 4 Figure 1: Pattern 1324 and its reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' narrowing expanding Figure 2: A 2-phase sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' since at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' We can thus remove those two elements, 4 now directly follows 1, and the persistence diagram of the reduced array plus the pair (2, 3) is equal to the persistence diagram of the original array, see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content='. After reducing the sequence based on those remarks, we are left with a sequence with no 123, 321, 1324 or 4231 pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' It is easy to see that such a sequence has a very specific 2-phase shape depicted in Figure 2: it alternates between local minima and local maxima, in a first expanding phase the maxima are increasing and the minima decreasing, and in a second narrowing phase the maxima are decreasing and the minima increasing (of course one of the phases may be empty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' Indeed, as soon as we see a pattern 132, the next element can be neither lower than 2 (pattern 321) nor higher than 3 (pattern 1324), so the triple that follows and shares 2 elements with 132 can only follow the pattern 312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' Similarly, 312 can only be followed by 132, and the narrowing pattern only stops at the end of the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' The expanding phase is symmetric to the narrowing one, and we only need to notice that 2 consecutive local minima must be in a pattern 132 or 231 to conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' The extremities also provide some opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' For instance if the sequence starts with 21, we can remove 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' If it starts with 231, we can pair off 2 and 3, remove them and start the sequence at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' Symmetric operations are possible at the other extremity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' This is sufficient to reduce a 2-phased sequence as obtained above to just one point, the global minimum, and we have the whole persistence diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' To apply these operations efficiently, we propose adding the values one by one from left to right, maintaining a reduced sequence with the invariant that it has a narrowing pattern and starts by 12 (unless it is reduced to a single 2 value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' This invariant is equivalent to the absence of patterns 123, 321, 1324 and 4231, and of patterns 21 and 231 at the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' With every new value, we check if appending it breaks the invariant, or equivalently if a simplification involving the new element is possible, and we simplify until the invariant is restored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' After processing the whole input, and after removing the last element in case the reduced sequence ends in 12, we can just pair and remove the last 2 elements (terminating 132 pattern) recursively until only 1 element remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' Since we only go back when removing elements, the amortized complexity per element is constant and the whole algorithm takes linear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' 2 Function on a circle For a PL function defined on the circle S1, we do not have extremities at which we could simplify, but that is unnecessary, since removing the patterns 123/321 and 1324/4231 is sufficient to get down to just 2 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' Indeed, without 123 and 321, the sequence has an even length and alternates between local minima and maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' If the sequence has length at least 4, 2 consecutive minima are involved in a pattern 132 or 231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' By symmetry, we assume there is a pattern 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' As in the line case, when a narrowing pattern starts, it cannot end until the end of the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' However, on a circle, the sequence is periodic and does not end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' In particular, it reaches this 1 again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' In a narrowing sequence, the minima increase, so when we reach 1 again, it must be larger than 2, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' From an algorithmic point of view, we can cut the circle at an arbitrary point, make a first pass to get a 2-phase pattern, then reconnect the 2 extremities and simplify at the junction until we have only 2 values left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' It may be convenient to use the global minimum (or maximum, or both) as initial cutting point, although it does not change the linear complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' 3 Parallelism Although we expect this approach is fast enough in practice not to require parallelization, it is tempting to try it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' We can split the segment into smaller segments, simplify each of them to a 2-phase shape in parallel, and collect the pairs the simplifications find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' And we can iterate, possibly using the point where the phase changes (the minimum) inside each segment as the new splitting points, or merging adjacent segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' For a fairly nice function, this should reduce the sequence significantly and let us finish sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' However, if for instance the input already has a narrowing shape from the beginning, the parallel phases will do almost nothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' This is then just a heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' There is little hope of a perfectly parallel algorithm because of the non-locality of persistent homology, Figure 3 shows that for a narrowing shape, adding at the end a very large or very small value can change the pairing of all the points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' References [1] The GUDHI Project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' GUDHI User and Reference Manual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' GUDHI Editorial Board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' URL: https://gudhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content='inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content='fr/doc/latest/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQf1Qur/content/2301.04745v1.pdf'} +page_content=' 3 Figure 3: Non-locality: the last element may determine the pairing of all the other points.' metadata={'source': 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Barkan and Jan Steinebrunner +January 23, 2023 +Abstract +We develop an analog of Dugger and Spivak’s necklace formula providing an explicit de- +scription of the Segal space generated by an arbitrary simplicial space. We apply this to obtain +a formula for the Segalification of 푛-fold simplicial spaces, a new proof of the invariance of right +fibrations, and a new construction of the Boardman–Vogt tensor product of ∞-operads for, which +we also derive an explicit formula. +Contents +1 +Segalification +5 +1.1 +Necklace contexts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +1.2 +Segalification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +1.3 +Variations on the Segalification formula . . . . . . . . . . . . . . . . . . . . . . . . . . +9 +2 +Applications +12 +2.1 +Segalification and right fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +12 +2.2 +Segalification for (∞,푛)-categories +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +2.3 +The Boardman-Vogt tensor product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +17 +References +20 +Introduction +A particularly useful and robust perspective on ∞-categories is provided by simplicial spaces. +Recall that a Segal space is a simplicial space 푋 : 횫op → S for which the natural map 푋푛 +≃−→ 푋1 ×푋0 +· · · ×푋0 푋1 is an equivalence for all 푛. A Segal space is complete if the map 푠0 : 푋0 → 푋1 induces an +equivalence onto a certain union of components 푋 eq +1 +⊂ 푋1. We let PshCSS(횫) ⊂ Pshseg(횫) ⊂ Psh(횫) +denote the full subcategories of (complete) Segal spaces. There are localizations: +PshCSS(횫) +Pshseg(횫) +Psh(횫) +L퐶 +Lseg +⊣ +⊣ +1 + +and we denote the composite left adjoint by LCSS : Psh(횫) → PshCSS(횫). The ∞-category Cat∞ of +∞-categories participates in an adjunction +C: Psh(횫) ⇄ Cat∞ :N• +where the left adjoint +C is left Kan extended from the inclusion 횫 ⊂ Cat∞. The right adjoint +N• is given by restricting the Yoneda embedding along 횫 ⊂ Cat∞ and induces an equivalence +N• : Cat∞ ≃ PshCSS(횫) under which +C(−) is identified with LCSS. To avoid confusion we will write +LCSS for the localization endofunctor of Psh(횫) and +C(−) for the left adjoint with values in Cat∞. +In his foundational work on complete Segal spaces [Rez01], Rezk provides a formula for the Rezk- +completion functor L퐶. The purpose of this note is to provide a formula for the “Segalification” +functor Lseg. Combining the two one obtains an explicit description of the ∞-category generated +by an arbitrary simplicical space. +Necklaces and Segalification. Our formula for Lseg is heavily influenced by the work of Dugger +and Spivak [DS11] on the rigidification of quasicategories. It will involve a colimit indexed by +a certain category of “necklaces” [DS11, §3] which we now recall. A necklace is a simplicial set +푁 = Δ푛1 ∨ · · · ∨ Δ푛푘 obtained by joining standard simplices at their start and endpoints, as indicated +in Fig. 1. Following Dugger and Spivak we define Nec to be the (non-full) subcategory of sSet +휆 +Figure 1: A morphism of necklaces 휆: Δ2 ∨ Δ2 ∨ Δ1 ∨ Δ1 → Δ1 ∨ Δ3 ∨ Δ2 defined by collapsing the +first edge and including the remaining necklace as indicated. +whose objects are necklaces and whose morphisms are maps of simplicial sets 푓 : Δ푛1 ∨ · · · ∨ Δ푛푘 → +Δ푚1 ∨ · · · ∨ Δ푚푙 that preserve the minimal and maximal elements. +We can now state the formula for Lseg in terms of necklaces. For the sake of simplicity we state +here the formula only for the case of 1-simplices Lseg(푋)1. This suffices to determine L(푋)푛 for all +푛 by the Segal condition. +Theorem A. For every simplicial space 푋 ∈ Psh(횫) there is a canonical equivalence: +LSeg(푋)1 ≃ colim +푁 ∈Necop MapPsh(횫) (푁, 푋) +≃ +colim +Δ푛1∨···∨Δ푛푟 ∈Necop 푋푛1 ×푋0 · · · ×푋0 푋푛푟 +One could in fact deduce this theorem from the results of Dugger–Spivak by passing through +the various model categories for ∞-categories, but we will instead give a “synthetic” proof as we +believe it to be insightful. +Application: right fibrations. Our first application is to the notion of right fibrations of simplicial +spaces in the sense of Rezk (see [Ras17, Remark 3.1]). Using the Segalification formula we show +that right fibrations of simplicial spaces are invariant under LCSS-equivalences, i.e. if 푓 : 푋 → 푌 is +2 + +a map of simplicial spaces such that LCSS(푓 ) is an equivalence then base change along 푓 induces +an equivalence 푓 ∗ : Psh(횫)r−fib +/푌 +≃ Psh(횫)r−fib +/푌 +. This implies that right fibrations over an arbitrary +simplicial space 푋 model presheaves on the associated ∞-category +C(푋): +Corollary B (Rasekh). For any simplicial space 푋 the functor +C(−) induces an equivalence +Psh(Δ)r−fib +/푋 +C(−) +−−−−→ +≃ +Catr−fib +∞/C(푋) ≃ Psh( +C(푋)) +A model-categorical version of this result was previously proven by Rasekh [Ras17, Theorem 4.18 +and 5.1]. Our proof has the advantage of being “synthetic” and also significantly shorter. An +alternative formulation of this corollary is to say that (the nerve of) the universal right fibration +Sop +∗ +→ Sop classifies right fibrations of arbitrary simplicial spaces.1 +We shall now use the above result to give a formula for +C(−) : Psh(횫) → Cat∞. Let us write +횫max ⊆ 횫 for the wide subcategory spanned by morphisms which preserve the maximal element, +and given a simplicial space 푋 : 횫op → S let us write 푝푋 : 횫/푋 → 횫 for the associated right fibration. +Corollary C. For every simplicial space 푋 ∈ Psh(횫), the last vertex functor 횫/푋 → +C(푋) induces an +equivalence of ∞-categories +횫/푋 [푊 −1 +푋 ] ≃ +C(푋) , +where 푊푋 ≔ 푝−1 +푋 (횫max) ⊆ 횫/푋. +In the case that 푋 is level-wise discrete, i.e. a simplicial set, this recovers a result of Stevenson +[Ste17, Theorem 3] who attributes it to Joyal [Joy07, §13.6]. +Application: (∞,푛)-categories. The ∞-category of (∞,푛)-categories admits many equivalent de- +scriptions including Rezk’s complete Segal Θ푛-spaces [Rez10] and Barwick’s complete 푛-fold Segal +spaces [Bar]. These were shown to be equivalent by Barwick–Schommer-Pries [BS21] and later us- +ing different techniques also by Bergner–Rezk [BR20] and Haugseng [Hau18]. We refer the reader +to [Hau21, §2] for a brief account of different models and some known comparisons between them. +We shall now present an application of our main result to 푛-fold Segal spaces. +We say that an 푛-fold simplicial space 푋 : (횫op)×푛 → S is reduced if each of the (푛 − 푘 − 1)-fold +simplicial spaces 푋푚1,...,푚푘,0,•,...,• is constant. We write Psh푟 (횫×푛) ⊆ Psh(횫×푛) for the full subcategory +of reduced 푛-fold simplicial spaces and let Seg푛−fold +횫op +⊆ Psh푟 (횫×푛) denote the full subcategory +spanned by the 푛-fold Segal spaces, i.e. those that satisfy the Segal condition in each coordinate. +This inclusion Seg푛−fold +횫op +↩→ Psh푟 (횫×푛) admits a left adjoint and we give a formula for it: +Theorem D. The left adjoint L: Psh푟 (횫×푛) → Seg푛−fold +횫op +may be computed as L = L푛 ◦ · · · ◦ L1 where +L푗 : Psh(횫×푛) → Psh(횫×푛) denotes the endofunctor that segalifies the 푗th coordinate: +(L푗푋)푚1,...,푚푗−1,1,푚푗+1,...,푚푛 ≃ +colim +Δ푛1∨···∨Δ푛푟 ∈Necop 푋푚1,...,푛1,...,푚푛 +× +푋푚1,...,0,...,푚푛 +· · · +× +푋푚1,...,0,...,푚푛 +푋푚1,...,푛푟,...,푚푛. +Note that here the order of the L푗 is crucial: we need to first Segalify 1-morphisms, then 2- +morphisms, etc. If one were to apply L2 and then L1 the result would not necessarily satisfy the +Segal condition in the second coordinate. +1Note that the content of this statement is not so much that it identifies the universal right fibration of simplicial spaces +(this can be done easily since the simplices Δ푛 are complete Segal spaces), but rather that it states that right fibrations of +simplicial spaces admit a universal example. +3 + +Application: the Boardmann–Vogt tensor product. One of the many great achievements of +Lurie’s book project on Higher Algebra [Lur] is the construction of a homotopy coherent symmetric +monoidal structure ⊗Lurie on the ∞-category of ∞-operads Op∞ generalizing the Boardmann-Vogt +tensor product [BV73, §II.3]. The defining property of O ⊗Lurie P is that algebras over it are “O- +algebras in P-algebras”, i.e. that for any symmetric monoidal ∞-category C there is an equivalence +AlgO⊗LurieP (C) ≃ AlgO(AlgP (C)). +The construction of ⊗Lurie is quite intricate as it involves a delicate mix of quasicategorical and +model categorical techniques. We shall now describe how the necklace formula can be used to +justify a simpler alternative approach to the tensor product of ∞-operads. +Theorem E. The tensor product of symmetric monoidal ∞-categories uniquely restricts to a tensor product +⊗BV on Op∞ such that the envelope Env: (Op∞, ⊗BV) → (Cat⊗ +∞, ⊗) is a symmetric monoidal functor. +Moreover, for any two ∞-operads O and P there is a canonical equivalence O ⊗BV P ≃ O ⊗Lurie P. +Remark. Note that Theorem E does not claim that (Op∞, ⊗BV) and (Op∞, ⊗Lurie) are equivalent as +symmetric monoidal ∞-categories. It does however reduce the question to whether the envelope +can be constructed as a symmetric monoidal functor (Op∞, ⊗Lurie) � (Cat⊗ +∞, ⊗). This is not entirely +clear since the higher coherence data (associator, braiding, etc. . . ) of Lurie’s tensor product ⊗Lurie +is tricky to access2 as it seemingly has no obvious universal property. Moreover, the authors are +unaware of any applications in which the specific coherence of Lurie’s construction plays a role. +A novel consequence of Theorem E is that, at least in principle, the Boardman-Vogt tensor product +is only as difficult to compute as necklace colimits. The resulting formula will be easiest to express +in the language of symmetric sequences. +Outlook: symmetric sequences. A symmetric sequence is a presheaf on the category of finite +sets and bijections. The disjoint union ⊔ and the product × of finite sets extend via Day convolu- +tion to symmetric monoidal structures on symmetric sequences SymSeq ≔ Psh(Fin�) which we +respectively denote by ⊗ and ⊠. Since (SymSeq, ⊗) is the free presentably symmetric monoidal +∞-category on a single generator 1 ∈ SymSeq, evaluation induces an equivalence +ev1 : FunCAlg(PrL) +�(SymSeq, ⊗), (SymSeq, ⊗)� +≃−→ SymSeq +which endows SymSeq with yet another (non-symmetric) monoidal structure ◦ coming from the +composition of endofunctors on the left side. It is generally expected that 1-colored (non-complete) +∞-operads are equivalent to associative algebras for ◦ in SymSeq, and for a different definition +of ◦ this was shown in [Hau22]. Given two such algebras O, P ∈ AlgE1(SymSeq, ◦) the necklace +formula in this setting gives: +O ⊗BV P ≃ +colim +Δ푛1∨···∨Δ푛푟 ∈Necop (O◦푛1 ⊠ P◦푛1) ◦ · · · ◦ (O◦푛푟 ⊠ P◦푛푟 ) +A formal proof of this does not fit within the scope of the present paper, as it relies on a good +interface between ∞-operads and symmetric sequences. We plan to provide such an interface +and provide a full proof of this in forthcoming work where we revisit the composition product of +symmetric sequences from the perspective of equifibered theory. +2Lurie does give a model categorical construction of a (non-symmetric) monoidal structure which does have a recog- +nizable universal property as a certain localization of ∞-categories over Fin. The symmetric monoidal structure however is +constructed by hand and apart from the binary operation the relation between the two is not commented on. +4 + +Acknowledgments +We would like to thank Rune Haugseng for helpful comments on an earlier draft and Maxime +Ramzi for useful conversations related to this paper. +The first author would like to thank the Hausdorff Research Institute for Mathematics for their +hospitality during the fall 2022 trimester program, funded by the Deutsche Forschungsgemein- +schaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2047/1 – +390685813. The second author is supported by the Danish National Research Foundation through +the Copenhagen Centre for Geometry and Topology (DNRF151). +1 +Segalification +1.1 +Necklace contexts +Let us fix an ∞-category C. In this section we study the general problem of approximating a +reflective localization functor L: Psh(C) → Psh(C) in the sense of [Lur09, Proposition 5.2.7.4] using +a suitable auxiliary subcategory of Psh(C). +Definition 1.1. A presheaf 푋 ∈ Psh(C) is called L-local if the canonical map 푋 → L(푋) is an +equivalence, i.e. if 푋 lies in the essential image of L. A morphism of presheaves 푓 : 푌 → 푍 ∈ Psh(C) +is called an L-local equivalence if L(푓 ) is an equivalence. +Definition 1.2. A necklace context is a triple (C, L, N ) where C and L are as above and N ⊆ Psh(C) +is a full subcategory such that +1. Y(푐) ≔ MapC (−,푐) is L-local for all 푐 ∈ C. +2. Y(푐) ∈ N for all 푐 ∈ C and L(푁) ∈ Y(C) for all 푁 ∈ N . +Example 1.3. A compatible necklace category for a pair (C, L) as in Definition 1.2 exists if and +only if the first condition holds. In this case, the minimal possible necklace category is given +by the representable presheaves Nmin ≔ Y(C) ⊆ Psh(C) and the maximal choice is given by +Nmax ≔ L−1(C) ⊆ Psh(C), namely all 푋 ∈ Psh(C) such that L(푋) ∈ Y(C). The full subcategory +Nsub ⊆ Psh(C) spanned by all subobjects 퐴 ⊆ Y(푐) such that L(퐴) ≃ Y(푐) is another possible choice. +Given a necklace context (C, L, N ), the Yoneda embedding Y : C ↩→ Psh(C) lands in N ⊆ Psh(C) +and thus gives rise to an adjunction +푙 ≔ L|N : N +C :Y =:푖 +⊣ +Passing to to presheaves we obtain a quadruple adjunction +Psh(C) +Psh(N ). +푖!=푙∗ +푖∗=푙∗ +푙! +푖∗ +⊣ +⊣ +⊣ +5 + +Lemma 1.4. The natural transformation 훽 : 푖∗ → 푙! defined by +� +훽 : 푖∗ 푖∗◦푢 +−−−→ 푖∗ ◦ (푙∗ ◦ 푙!) ≃ (푙 ◦ 푖)∗ ◦ 푙! ≃ 푙! +� +∈ Fun(Psh(N ), Psh(C)). +is L-local. +Proof. The functors L ◦ 푖∗ and L ◦ 푙! are both left adjoints, when thought of as functors Psh(N ) −→ +PshL−loc(C). +Therefore it will suffice to check that L(훽푍) is an equivalence for representable +presheaves 푍 = Y푁 with 푁 ∈ N , as these generate Psh(N ) under colimits. Restricting Y푁 along 푖 +we obtain 푖∗Y푁 = 푁 ∈ Psh(C). One the other hand we have 푙!Y푁 ≃ Y푙 (푁) ≃ L(푁) and the natural +transformation in this case is the canonical map 훽Y푁 : 푁 → L(푁) which is indeed L-local. +□ +Definition 1.5. Given a necklace category (C, L, N ) we define +푄N : Psh(C) +푖∗−→ Psh(N ) +푙!−→ Psh(C). +This functor receives a canonical natural transformation from the identity +휆 : IdPsh(C) +≃ +←− 푖∗ ◦ 푖∗ +훽◦푖∗ +−−−→ 푙! ◦ 푖∗ = 푄N . +Remark 1.6. The functor 푖∗ : Psh(C) → Psh(N ) may be computed as (푖∗푋)(푁) ≃ MapPsh(C) (푁,푋). +By the pointwise formula for left Kan extensions thus have +푄N (푋)(푐) = +colim +(푁,푙 (푁)←푐) ∈(N ×C C푐/)op MapPsh(C) (푁, 푋). +Note that by Lemma 1.4, 휆: id → 푄N (푋) is L-local and thus the unit transformation id → L factors +through 휆. The resulting natural transformation 푄N → L is then L-local by cancellation. We thus +conclude: +Corollary 1.7. There exists a canonical L-local natural transformation 푄N → L such that for any 푋 ∈ +Psh(C) the map 푄N (푋) → L(푋) is an equivalence if and only if 푄N (푋) is L-local. +Remark 1.8. Since 푄N is accessible and 휆: id → 푄N satisfies QN ◦ 휆 ≃ 휆QN : QN → Q2 +N , we can +iterate 푄N transfinitely (in analogy with how sheafification is constructed in [Lur09, Proposition +6.2.2.7]) to obtain a localization. This localization will be equivalent to L if and only the morphisms +{푁 → L(푁)}푁 ∈N generate the class of L-local equivalences. +1.2 +Segalification +We now specialize to the setting of Segal spaces, where the localization Lseg is defined as the left +adjoint to the full inclusion +Seg횫op (S) ↩→ Fun(횫op, S) +of those simplicial spaces 푋• for which the map 푋푛 → 푋1 ×푋0 · · · ×푋0 푋1 is an equivalence. We will +choose a necklace category and show that 푄N ≃ Lseg. +6 + +Remark 1.9. The formula we will arrive at for Lseg is closely related to the work of Dugger–Spivak +[DS11], who construct a functor +ℭnec : sSet → sCat +from the category of simplicial sets to the category of simplicial categories, which they show to +be weakly equivalent to the left adjoint ℭ of the coherent nerve. This gives a formula for the +mapping spaces in ℭ(푍•) (as a colimit over the necklace category) when 푍• is a simplicial set in the +Joyal model structure. The case of a simplicial space follows by using the left Quillen equivalence +푡! : ssSet → sSet constructed by Joyal–Tierney [JT06]. +Segal spaces. Let us recall the category of necklaces, introduced by Dugger and Spivak [DS11]. +Definition 1.10. The concatenation 퐴 ∨ 퐵 of two bi-pointed simplicial sets (퐴,푎min,푎max) and +(퐵,푏min,푏max) is defined as the pushout +Δ0 +퐵 +퐴 +퐴 ∨ 퐵, +푎max +푏min +⌟ +which we point as (퐴 ∨ 퐵,푎min,푏max). This defines a (non-symmetric) monoidal structure on the +category of bi-pointed simplicial sets. +Definition 1.11. A necklace is a bi-pointed simplicial set obtained by concatenating simplices +(Δ푛, 0,푛), i.e. it is of the form 푁 = Δ푛1 ∨ · · · ∨ Δ푛푘. We let Nec denote the category whose objects are +necklaces and whose morphisms are maps of bi-pointed simplicial sets. +While the category Nec will play the central role in the Segalification formula, we will need a +slightly bigger category to set up the necklace context. +Definition 1.12. Let N ⊂ Psh(횫) denote the essential image of the faithful functor Nec → Psh(횫). +Lemma 1.13. Segalification Lseg : Psh(횫) → Seg횫op (S) restricts to a functor Lseg|N : N → 횫. +In +particular the triple (횫, Lseg, N ) is a necklace context. +Proof. We will show by induction on 푛 that the inclusion Δ{0,...,푛1 } ∨ · · · ∨ Δ{푛푘,...,푛} ↩→ Δ푛 is a Segal +equivalence thereby proving the claim. Consider the nested inclusion +Δ{0,1} ∨ · · · ∨ Δ{푛−1,푛} ↩→ Δ{0,...,푛1 } ∨ · · · ∨ Δ{푛푘,...,푛} ↩→ Δ푛 +The composite is a Segal equivalence by definition and since L preserves colimits and 푛푗+1 − 푛푗 < 푛 +the first map is a Segal equivalence by the induction hypothesis. The second map is therefore a +Segal equivalence by cancellation. +□ +Remark 1.14. Note that the map 푁 → Lseg(푁) = Δ푛 is a monomorphism for each necklace 푁. In +particular, for any two necklaces 푁, 푀, the map +MapN (푁, 푀) −→ MapPsh(횫) (Lseg(푁), Lseg(푀)) ≃ Map횫([푛], [푚]) +is a monomorphism, i.e. L: N → 횫 is faithful. +7 + +The Segal condition. Since (횫, Lseg, N ) is a necklace context we have by Lemma 1.13 a functor +푄 : Psh(횫) +푖∗−→ Psh(N ) +푙!−→ Psh(횫). +By Corollary 1.7 this comes with a Lseg-local natural transformation 푄 → Lseg. We may compute +the functor 푄(−) using Remark 1.6 as: +푄(푋)푛 = +colim +(푁,푙 (푁)←[푛]) ∈(N ×횫횫[푛]/)op MapPsh(횫) (푁, 푋). +Below we show that 푄(푋) is always a Segal space and thus by Corollary 1.7 that 푄 ≃ Lseg. +Definition 1.15. For any necklace 푁 we let 휄푁 : Δ1 → Lseg(푁) denote the unique map that preserves +the extrema. Given [푛] ∈ 횫 we define the functor +퐽 : +푛 +� +푖=1 +Nec ↩→ N ×횫 횫[푛]/, +by joining necklaces at their endpoints +(푀1, . . . , 푀푛) ↦−→ +� +푀1 ∨ · · · ∨ 푀푛, [푛] +Lseg (휄1∨···∨휄푛) +−−−−−−−−−−−→ Lseg(Lseg(푀1) ∨ · · · ∨ Lseg(푀푛)) ≃ Lseg(푀1 ∨ · · · ∨ 푀푛) +� +. +Lemma 1.16. The functor 퐽 is fully faithful and admits a right adjoint. In particular it is coinitial. +Proof. Fully-faithfulness follows by unwinding definitions. We claim that a right adjoint to 퐽 is +given by the following +퐽 푅 : (푁, 훼 : [푛] → Lseg(푁)) ↦−→ (푁훼 (0),훼 (1), . . . , 푁훼 (푛−1),훼 (푛)) +where 푁훼 (푗),훼 (푗+1) ≔ 푁 ∩ Δ{훼 (푗),...,훼 (푗+1) }. To see this, note that a tuple of necklace morphisms +(푀1, . . . , 푀푛) → (푁훼 (0),훼 (1), . . . , 푁훼 (푛−1),훼 (푛)) = 퐽 푅(푁, 훼) +is equivalent to a morphism 푀1 ∨ · · · ∨ 푀푛 → 푁훼 (0),훼 (1) ∨ · · · ∨ 푁훼 (푛−1),훼 (푛) ⊂ 푁 such that 푀푖 lands +in 푁훼 (푖−1),훼 (푖). +These can be identified with morphisms 퐽 (푀1, . . . , 푀푛) → (푁,훼) in N ×횫 횫[푛]/. +Therefore 퐽 푅 is indeed right adjoint to 퐽. +□ +Observation 1.17. The cofinality in Lemma 1.16 implies that 푄(푋)푛 may be computed as: +푄(푋)푛 ≃ +colim +(푀1,...,푀푛) ∈(Necop)푛 MapPsh(횫) (푀1 ∨ · · · ∨ 푀푛,푋) +≃ +colim +(푀1,...,푀푛) ∈(Necop)푛 MapPsh(횫) (푀1, 푋) × +푋0 . . . × +푋0 MapPsh(횫) (푀푛, 푋) +In particular for 푛 = 0 we just get 푄(푋)0 = 푋0. While this is a simplification of the general formula +from Remark 1.6, it has the downside that the functoriality in [푛] is not clear in general. However, +we can still see the functoriality in inert maps 휑 : [푚] ֌ [푛] as it is simply given by restricting to the +푀푖 that correspond to the image of 휑. This functoriality will suffice to check the Segal condition. +Proposition 1.18. For any simplicial space 푋• the simplicial space 푄(푋)• is a Segal space. +8 + +Proof. Consider the following diagram: +colim +(푀1,...,푀푛) ∈(Necop)푛 Map(푀1,푋) × +푋0 . . . × +푋0 Map(푀푛,푋) +푄(푋)푛 +colim +푀1 ∈Necop Map(푀1, 푋) × +푋0 . . . × +푋0 colim +푀푛 ∈Necop Map(푀1,푋) +푄(푋)1 +× +푄 (푋)0 +. . . +× +푄 (푋)0 +푄(푋)1. +The horizontal maps are equivalences by Lemma 1.16 and Observation 1.17. The left vertical map +is an equivalence since the cartesian product in S/푋0 preserves colimits in each variable. +□ +1.3 +Variations on the Segalification formula +A formula for mapping spaces. Given a Segal space 푋 ∈ Seg횫op(S) the mapping spaces in the +associated ∞-category +C(푋) may be computed as +Map +C(푋) (푥,푦) ≃ {푥} ×푋0 푋1 ×푋0 {푦} +for any 푥,푦 ∈ 푋0. Below we show how to use the results of the previous to derive a formula for +these mapping spaces when 푋 is an arbitrary simplicial space. In the case where 푋 is a simplicial +set this recovers the formula of Dugger–Spivak [DS11], which inspired our Segalification formula. +Lemma 1.19. For any simplicial space 푋 ∈ Psh(횫) and 푥,푦 ∈ 푋 there is a canonical equivalence +|Nec/(푋,푥,푦) | ≃ Map +C(푋) (푥,푦). +where Nec/(푋,푥,푦) ⊆ +� +Psh(횫)Δ0⊔Δ0/ +� +/(푋,푥,푦) denotes the full subcategory spanned by necklaces. +Proof. Interpreting Nec as a full subcategory of Psh(횫)Δ0⊔Δ0/ by recording the minimal and maximal +vertex we can fit Nec/(푋,푥,푦) into a cartesian square: +Nec/(푋,푥,푦) +Nec ×Psh(횫) Psh(횫)/푋 +{(푥,푦)} +푋0 × 푋0 +The top right corner is a right fibration over Nec corresponding to the presheaf 푖∗(푋) : Necop → S. +Its weak homotopy type is thus the colimit of this functor, which is precisely the definition of 푄(푋)1. +The bottom edge in the square is a map of spaces, hence inverting all maps we obtain +|Nec/(푋,푥,푦) | ≃ {(푥,푦)} ×푋 ×2 +0 푄(푋)1 ≃ Map +C(푋) (푥,푦), +where the second equivalence holds since the Rezk-completion of 푄(푋) ≃ Lseg(푋) is the nerve of +C(푋). +□ +Remark 1.20. Given three points 푥,푦, 푧 ∈ 푋 the monoidal structure ∨ on Nec yields a functor +∨: Nec/(푋,푥,푦) × Nec/(푋,푦,푧) −→ Nec/(푋,푥,푧). +Onweakhomotopytypesthisyieldsthecomposition Map +C(푋) (푥,푦)×Map +C(푋) (푦,푧) → Map +C(푋) (푥,푧) +in +C(푋). (As in [DS11, Eqn. 1.2].) This can be seen by an argument similar to Lemma 1.19 to the +necklace formula for 푄2푋. +9 + +1-categories. Similar to how Δop-colimits in 1-categories can be computed as reflexive coequalizers, +Necop colimits in a 1-category can be reduced to certain “thin” necklaces. +Definition 1.21. We say that a necklace 푁 = Δ푛1 ∨ . . . Δ푛푟 ∈ Nec is thin if � +푖 푛푖 ≤ 푟 + 1 and 푛푖 ≥ 1, +in other words if it consists of 1-simplices and at most one two-simplex. If 푁 consists only of +1-simplices we say that it is very thin. +Let Necthin ⊂ Nec denote the full subcategory of thin +necklaces. +Lemma 1.22. The full inclusion Necop +thin ↩→ Necop is 1-cofinal, that is, for any functor Necop → C to a +1-category C the colimit may equivalently be computed over Necop +thin. +Proof. We need to show that for any necklace 푁 = Δ푛1 ∨ · · · ∨ Δ푛푟 ∈ Nec the slice category Necthin/푁 +is connected. We enumerate the vertices of 푁 in their canonical order as 0, . . . ,푛 = � +푖 푛푖. A very +thin necklace over 푁, Δ1 ∨ · · · ∨ Δ1 → 푁 may equivalently be encoded as a non-decreasing path +0 = 푎0 ≤ · · · ≤ 푎푘 = 푛 in [푛]. These paths are subject to the condition that we never have strict +inequalities 푎푙 < �푠 +푖=1 푛푖 < 푎푙+1 for any푠 and푙. Suppose that 푝 = (0 = 푎0 ≤ · · · ≤ 푎푘 = 푛) is such a path +and 푠 is such that 푝′ = (0 = 푎0 ≤ . . . �푎푠 · · · ≤ 푎푘 = 푛) is still an admissible path. Then there is a thin +necklace 푀 with a two-simplex (푎푠−1 ≤ 푎푠 ≤ 푎푠+1) that contains both of these paths. In particular, +the paths are connected through a zig-zag 푝 → 푀 ← 푝′ as objects of Necthin/푁 . Proceeding be +removing a vertex whenever possible we see that every very thin necklace over 푁 is connected in +Necthin/푁 to a very thin necklace that corresponds to a minimal path in 푁. But there is only one +path in 푁 that is minimal with respect to removing vertices, namely (0 ≤ 푛1 ≤ · · · ≤ �푟−1 +푖=1 푛푖 ≤ 푛). +Therefore all the very thin objects in Necthin/푁 are connected, and thus the category is connected +as every (thin) necklace contains a very thin necklace. +□ +Example 1.23. Suppose that푋• is a simplicial space and we want to compute the homotopy category +ℎ1( +C(푋)). For simplicity, let us assume that 푋푛 is discrete for all 푛.3 Then the set of morphisms in +ℎ1( +C(푋))) is exactly 휋0(Lseg(푋)1) and we may compute it as the colimit +Mor(ℎ1( +C(푋))) � colim +푁 ∈Necop Map(푁, 푋) � +colim +Δ푛1∨···∨Δ푛푟 ∈Necop 푋 (Δ푛1) ×푋 (Δ0) · · · ×푋 (Δ0) 푋 (Δ푛푟 ). +in the 1-category of sets. By Lemma 1.22 it suffices to take the colimit over Necop +thin. Moreover the +very thin necklaces are 0-cofinal, so the colimit may be expressed as a coproduct over the very thin +necklaces modulo an equivalence relation. This leads to the formula +Mor(ℎ1( +C(푋))) � +�� +푛≥0 +푋1 ×푋0 · · · ×푋0 푋1 +� +/∼ +where the equivalence relation is generated by (푓1, . . . , 푓푛) ∼ (푓1, . . . , 푓푖−1,푔, 푓푖+2, . . . , 푓푛) whenever +there is a 2-simplex in 푋 witnessing 푓푖+1 ◦ 푓푖 = 푔, and (푓1, . . . , 푓푛) ∼ (푓1, . . . , 푓푖−1, 푓푖+1, . . . , 푓푛) whenever +푓푖 is a degenerate 1-simplex. This recovers the classical formula for the homotopy category of a +simplicial set: namely, it is the free category on the edges of 푋 modulo the relations generated by +the 2-simplices. +3This is not a very restrictive assumption. Starting with a general simplicial space 푌• we may base change it along a +휋0-surjective map 푍0 → 푌0 to get a simplicial space 푍푛 = (푍0)푛+1 ×푌푛+1 +0 +푌푛 such that the resulting functor +C(푍•) → +C(푌•) +will be an equivalence. Now we may choose 푍0 to be discrete and define 푋푛 ≔ 휋0(푍푛). Then +C(푍•) → +C(푋•) induces an +equivalence on homotopy categories. +10 + +Segalificationin other categories. In this section we establish criteria on a presentable ∞-category, +which guarantee that Segalification is given by the necklace formula. This is summarized by the +following result, which we prove in the remainder of this section. +Proposition 1.24. Let V be a presentable ∞-category in which sifted colimits are stable under base change. +Then the left adjoint to the inclusion Seg횫op (V) ↩→ Fun(횫op, V) is given by the necklace formula: +Lseg(푋)1 ≃ colim +푁 ∈Necop(푖∗푋)(푁) ≃ +colim +Δ푛1∨···∨Δ푛푘 ∈Necop 푋푛1 ×푋0 · · · ×푋0 푋푛푘. +where 푖∗ denotes the right Kan extension 푖∗ : Fun(횫op, V) → Fun(N op, V). +Recall that if 픛 is an ∞-topos then all colimits in 픛 are universal [Lur09, Proposition 6.1.3.19]. In +particular Proposition 1.24 applies to ∞-topoi. A wider variety of examples is provided by passing +to algebras over ∞-operads. +Example 1.25. Let V be a presentably symmetric monoidal ∞-category4 and O be an ∞-operad. +Then the forgetful functor AlgO(V) → V creates and preserves both limits and sifted colimits [Lur, +3.2.2.4 & 3.2.3.1]. Consequently, if sifted colimits in V are stable under base change, then the same +holds for AlgO(V). +Lemma 1.26. Proposition 1.24 holds if we assume that Necop-colimits in V are stable under base change. +Proof. The left adjoint Lseg exists by the adjoint functor theorem. Since V is presentable we may +find a small ∞-category E and a fully faithful right adjoint 퐼 : V ↩→ Psh(E ). We denote the resulting +adjunction on presheaf categories by +퐼 횫 : Fun(횫op, V) ⇄ Fun(횫op, Psh(E )) :퐿횫 +We now define an endofunctor 푄V ≔ 퐿횫 ◦ 푄 ◦ 퐼 횫 : Fun(횫op, V) → Fun(횫op, V) where 푄 is the +endofunctor on Fun(횫op, Psh(E )) ≃ Fun(E , Psh(횫)) given pointwise given by the usual formula +(see Observation 1.17). This 푄V receives a natural transformation +휆V : Id ≃ 퐿횫 ◦ 퐼 횫 퐿횫◦휆◦퐼횫 +−−−−−−→ 퐿횫 ◦ 푄 ◦ 퐼 횫 = 푄V +coming from 휆: id → 푄. This transformation is a Segal equivalence. Indeed, if 푋,푌 : 횫op → V and +푌 is Segal, then in the commutative square +MapFun(횫op,V)(푄V푋,푌) +MapFun(횫op,V) ((퐿횫 ◦ 퐼 횫)(푋),푌) +MapFun(횫op,V)((푄 ◦ 퐼 횫)(푋), 퐼 횫(푌)) +MapFun(횫op,V) (퐼 횫(푋), 퐼 횫(푌)) +≃ +≃ +(−)◦휆V +(−)◦휆 +the bottom map is an equivalence since 퐼 횫(푌) is Segal and thus so is the top map. +It remains to show that 푄V (푋) is Segal for all 푋 : 횫op → V. This follows from the same proof as +Proposition 1.18 by using that Necop-colimits are stable under base change. +□ +4In fact it suffices to ask that the monoidal structure is compatible with sifted colimits. +11 + +Diagrams of shape Necop may be difficult to recognize in the wild. Fortunately, Necop is a sifted +category, colimits over which are well understood. We will deduce this from the following fact to +which it is intimately linked: +Lemma 1.27. The Segalification functor L: Psh(Δ) → Psh(Δ) preserves products. +Proof. The two functors +Psh(Δ) × Psh(Δ) −→ Cat∞ +given by (푋,푌) ↦→ L(푋 × 푌) and L(푋) × L(푌) both preserve colimits in both variables. Therefore it +suffices to check that the natural transformation between them is an equivalence on (Δ푛, Δ푚). But +in this case it is easy to check because Δ푛, Δ푚 and Δ푛 × Δ푚 are all Segal spaces. +□ +Lemma 1.28. The category Necop is sifted. +Proof. We need to show that the diagonal functor Δ: Necop → Necop×Necop is cofinal. Equivalently, +we need that for all 퐴, 퐵 ∈ Nec the slice Nec ×Nec2 Nec2 +/(퐴,퐵) is weakly contractible. This category is +equivalent to the full subcategory Nec/퐴×퐵 ⊆ +� +Psh(횫)Δ0⊔Δ0/ +� +/퐴×퐵 spanned by necklaces, where the +product퐴×퐵 is taken in the ∞-category Psh(횫)Δ0⊔Δ0/ of bipointed simplicial spaces. By Lemma 1.19 +the weak homotopy type of this category computes the mapping space +|Nec/(퐴×퐵,(푎min,푏min),(푎max,푏max))| ≃ Map +C(퐴×퐵) ((푎min,푏min), (푎max,푏max)). +Since +C(−) commutes with products by Lemma 1.27, we may compute +C(퐴 × 퐵) ≃ +C(퐴) × +C(퐵) = +[푛] × [푚]. In particular we see that the mapping space Map[푛]×[푚]((0, 0), (푛,푚)) is contractible, +which completes the proof. +□ +2 +Applications +2.1 +Segalification and right fibrations +Throughout this section we fix a presentable ∞-category V and a factorization system (V퐿, V푅). +Definition 2.1. We say that 푋 : Δop → V is right-V푅-fibered if 푑0 : 푋푛 → 푋푛−1 is in V푅 for all 푛 ≥ 1. +Observation 2.2. A Segal object 푋 : 횫op → V is right-V푅-fibered if and only if 푑0 : 푋1 → 푋0 is in V푅. +Indeed morphisms in V푅 are closed under pullbacks in the arrow category and when 푋 is Segal +we can write 푑0 : 푋푛 → 푋푛−1 as a pullback in the arrow category of the following cospan +푋푛−1 +푋0 +푋1 +푋푛−1 +푋0 +푋0. +푑1◦···◦푑푛 += += +푑0 += +푑1◦···◦푑푛 +푑0 +Under suitable assumptions the necklace formula can be used to show that Segalification preserves +right-V푅-fibered objects. +12 + +Proposition 2.3. Suppose V and (V퐿, V푅) are such that: +1. sifted colimits in V are stable under base change and +2. V푅 +/푋 ⊆ V/푋 is closed under sifted colimits for all 푋 ∈ V. +Then Segalification Lseg : Fun(횫op, V) → Seg횫op (V) preserves right-V푅-fibered objects. +Proof. By Observation 2.2 it suffices to show that if 푋 : 횫op → V is right-V푅-fibered then 푑0 : +L(푋)1 → L(푋)0 is in V푅. We claim that for every necklace 푁 = Δ푛1 ∨ · · · ∨ Δ푛푘 the map (푖∗푋)(푁) ≃ +푋푛1 ×푋0 · · · ×푋0 푋푛푘 → 푋0 induced by the inclusion of the terminal vertex Δ0 → 푁 is in V푅. Indeed, +when 푁 is a simplex this holds by assumption and the general case follows by taking pullbacks +since morphisms in V푅 are closed under pullbacks in the arrow category. +Using the necklace formula (see Proposition 1.24), we can write 푑0 : L(푋)1 → L(푋)0 as a colimit +L(푋)1 ≃ colim +푁 ∈Nec (푖∗푋)(푁) −→ 푋0 = L(푋)0 +in V/푋0, of a diagram indexed by Necop, of morphisms (푖∗푋)(푁) → 푋0 that lie in V푅 +/푋0. Since Necop is +sifted (see Lemma 1.28) and V푅 +/푋0 ⊆ V/푋0 is closed under sifted colimits, the colimit lies in V푅 +/푋0. +□ +Right fibrations of simplicial spaces. We shall now apply Proposition 2.3 to right fibrations of +simplicial spaces in the sense of Rezk whose definition we briefly recall. +Definition 2.4. A map of simplicial spaces 푝 : 푋• → 푌• is called a right fibration if the square +푋푛 +푋푛−1 +푌푛 +푌푛−1 +푑0 +푝 +푝 +푑0 +is cartesian for all 푛 ≥ 1. +The Segalification formula implies the following: +Corollary 2.5. If 푝 : 푋 → 푌 is a right fibration, then so is the Segalification L(푝) : L(푋) → L(푌). +Proof. The target map 푡 : Ar(S) → S is a cartesian fibration and thus we have a factorization system +on Ar(S) whose right part Ar(S)cart ⊆ Ar(S) consist of the cartesian edges, equivalently pullback +squares. (The left part consists of morphisms which induce equivalence on the target.) Note that +푝 : 푋 → 푌 ∈ Ar(Psh(횫)) = Fun(횫op, Ar(S)) is right-Ar(S)cart-fibered if and only if it is a right +fibration, hence it suffices to verify the conditions of Proposition 2.3 for V = Ar(S) equipped with +the aforementioned factorization system. The first condition holds since colimits in Ar(S) are +universal. The second condition holds since colimits in S are stable under base change. +□ +Remark 2.6. Examination of the proof of Corollary 2.5 shows that the same result holds if S is +replaced with any presentable ∞-category V in which sifted colimits are stable under base change. +13 + +Recall that the nerve N• : Cat∞ → Psh(횫) is fully faithful and its essential image is precisely +the complete Segal spaces. We write +C(−) : Psh(횫) → Cat∞ for the left adjoint of N•. We let +LCSS: Psh(횫) → Psh(횫) denote the localization onto the complete Segal spaces. With this notation +we have for any 푋 ∈ Psh(횫) a canonical equivalence N• +C(푋) ≃ LCSS푋. A model-categorical proof +of the following proposition was given by Rasekh [Ras17, Theorem 4.18 and 5.1]. +Proposition 2.7. The functor +C: Psh(횫) → Cat∞ induces for any simplicial space 푋• an equivalence +Psh(횫)r−fib +/푋 +C(−) +−−−−→ +≃ +Catr−fib +∞/C(푋) ≃ Psh( +C(푋)) +Proof. Under the equivalence N• : Cat∞ ≃ CSeg횫op(S) the functor +C(−) is identified with the +localization LCSS. By Corollary 2.5 and [HK22, Proposition A.21] the Segalification functor Lseg +and the Rezk-completion functor L퐶 preserve right fibrations, so we have well-defined functors: +LLCC : Psh(횫)r−fib +/푋 +Lseg +−−−→ Psh(횫)r−fib +/Lseg푋 +L퐶 +−−→ Psh(횫)r−fib +/LCSS푋 +The second functor is an equivalence by [HK22, Proposition A.22] with inverse given by pullback +along Lseg푋 → LCSS푋. It thus remains to show that the first functor is an equivalence. Pulling back +along 휑푋 : 푋 → Lseg푋 defines functor in the other direction: +휑∗ : Psh(횫)r−fib +/푋 +−→ Psh(횫)r−fib +/Lseg푋 +Note that a priori it is unclear whether Lseg and 휑∗ are adjoints of one another. Nevertheless, we +will prove they are inverse by showing that the composites in both directions are equivalent to the +identity. +For the composite 휑∗ ◦ Lseg we start with a right fibration of simplicial spaces 푝 : 푋• → 푌• and +we need to show that the comparison map 푋• → (푌• ×Lseg푌• LCSS푋•) is an equivalence. Since this +is a map of right fibrations over LCSS푌•, it suffices by Lemma 2.8 that this is an equivalence on +0-simplices. But on 0-simplices we have (Lseg푋)0 ≃ 푋0 and (Lseg푌)0 ≃ 푌0, so the claim follows. +For the composite Lseg ◦ 휑∗ we start with a right fibration 푝 : 퐸• → Lseg푌• over Lseg푌• and we need +to show that the canonical dashed map in the diagram +휑∗ +푌퐸• +Lseg(휑∗ +푌 퐸•) +퐸• +푌• +Lseg푌• +is an equivalence. Again, this is a map of right fibrations over Lseg푌• and by Lemma 2.8 it may be +checked on 0-simplices. Using 푋0 ≃ (Lseg푋)0 twice we see that all horizontal maps in this diagram +are equivalences, in particular the dashed one. This completes the proof. +□ +Lemma 2.8. Suppose 푋• → 푌• and 푋 ′ +• → 푌• are right fibrations and 푓 : 푋• → 푋 ′ +• is a map over 푌•. Then 푓 +is an equivalence if and only if 푓0 : 푋0 → 푋 ′ +0 is. +Proof. Since 푓 : 푋 ′ +• → 푋• is a map of right fibrations the map 푓푛 : 푋푛 → 푋 ′ +푛 can be recovered as the +base change of 푓0 : 푋0 → 푋 ′ +0 along (푑0)푛 : 푋 ′ +푛 → 푋 ′ +0. +□ +14 + +A formula for +C(−). Let 푋 be a simplicial space and write 횫/푋 for its ∞-category of simplices, +i.e. the codomain of the associated right fibration 푝푋 : 횫/푋 → 횫. Proposition 2.7 can be used to +give a formula for +C(푋) as a certain localization of 횫/푋. To do so we will need the “last vertex +map” N•(횫/푋 ) → 푋 (see e.g. [HK22, §4]) and the “last vertex functor” 푒 : 횫/푋 → +C(푋) obtained +by applying +C(−) to it. Let us write 횫max ⊆ 횫 for the wide subcategory spanned by morphisms +which preserve the maximal element. +Corollary 2.9. The "last vertex functor" 푒 : 횫/푋 → +C(푋) induces an equivalence of ∞-categories +횫/푋 [푊 −1 +푋 ] ≃ +C(푋) . +where 푊푋 ≔ 푝−1 +푋 (횫max) ⊆ 횫/푋. +Proof. The functor 푒 factors through 푒 : 횫/푋 [푊 −1 +푋 ] → +C(푋). Pulling back along 푒 induces functors +Psh( +C(푋)) −→ Psh(횫/푋 [푊 −1 +푋 ]) ↩→ Psh(횫/푋 ) ≃ Psh(횫)/푋 +and by Proposition 2.7 the composite is fully faithful with image the right fibrations over 푋. By +[BS22, Lemma 4.1.8] the composite functor and the second functor have the same essential image +and since the latter is fully faithful the first functor is an equivalence. By naturality of the Yoneda +embedding it follows that 푒 is fully faithful. It remains to verify that 푒 : 횫/푋 → +C(푋) is essentially +surjective and indeed the composite +푋0 → (횫/푋)≃ → +C(푋)≃ ≃ (LCSS푋)0 +is surjective on components. +□ +2.2 +Segalification for (∞,푛)-categories +By iterating the Segalification formula one can also obtain formulas for the Segalification for +(∞,푛)-categories. We recall the definition of 푛-fold Segal spaces due to Barwick [Bar]. See [CS19, +Definition 2.2 and 2.4] and [Hau18] for a reference. +Definition 2.10. An 푛-fold simplicial space 푋 : 횫op,푛 → S is +• reduced if for every 푘 ≥ 0 and푚1, . . . ,푚푘 ∈ N the (푛−푘 −1)-fold simplicial space 푋푚1,...,푚푘,0,•,...,• +is constant. We denote by Psh푟 (횫×푛) the full subcategory spanned by reduced objects. +• an 푛-uple Segal space if it is Segal in each coordinate, that is, if for every 푘 ≥ 0 and +푚1, . . . ,푚푘−1,푚푘+1, . . . ,푚푛 ∈ N the simplicial space 푋푚1,...,푚푘−1,•,푚푘+1,...,푚푛 is a Segal space. +• an 푛-fold Segal space if it is an 푛-tuple Segal space and reduced. We denote by Segn−fold +횫op +⊂ +Psh푟 (횫×푛) the full subcategory spanned by the 푛-fold Segal spaces. +As we briefly explained in the introduction, complete 푛-fold Segal spaces model (∞,푛)-categories. +We will not discuss the issue of completeness in this paper, but rather our goal is it give a formula +for the Segalification of a reduced 푛-fold simplicial spaces. +Given 1 ≤ 푗 ≤ 푛 we denote by +L푗 : Psh(횫×푛) → Psh(횫×푛) the Segalification functor in the 푗-th coordinate. +15 + +Lemma 2.11. Suppose 퐹 : 퐾 → Psh(횫) is a diagram of simplicial spaces such that 퐾 is sifted, each 퐹 (푘) is +a Segal space, and the diagram 퐹 (−)0 : 퐾 → S is constant. Then the colimit colim +푘 ∈퐾 +퐹 (푘) is a Segal space. +Proof. A simplicial space 푋• is Segal if and only if the canonical map 푋푛 → 푋0 ×(푋0×푋0) (푋푛−1 × 푋1) +is an equivalence. In the case of 푋• = colim 푘 ∈퐾 퐹 (푘)• we therefore want show that the outside +rectangle in the following diagram is cartesian: +colim +푘 ∈퐾 +퐹 (푘)푛 +colim +푘 ∈퐾 +퐹 (푘)푛−1 × 퐹 (푘)1 +colim +푘 ∈퐾 +퐹 (푘)푛−1 × colim +푘 ∈퐾 +퐹 (푘)1 +colim +푘 ∈퐾 +퐹 (푘)0 +colim +푘 ∈퐾 +퐹 (푘)0 × 퐹 (푘)0 +colim +푘 ∈퐾 +퐹 (푘)0 × colim +푘 ∈퐾 +퐹 (푘)0 +≃ +Δ +≃ +The right horizontal maps are equivalences because 퐾 is sifted and hence it suffices to consider the +left square. This square is a colimit of the cartesian squares that we have because 퐹 (푘)• is Segal +for all 푘. Moreover, the bottom row of the squares is a constant functor in 푘 by assumption. So +it follows that the colimit of the square is still cartesian because colimits in S are universal. As +observed above this implies that colim 푘 ∈퐾 퐹 (푘)• is Segal as claimed. +□ +Lemma 2.12. Let 푋•,···• be a reduced 푛-fold simplicial space, then +1. (L푗푋)•,...,• is reduced for all 푗, and +2. if 푋•,...,• satisfies the Segal condition in the first (푗 − 1)-coordinates, then (L푗푋)•,...,• satisfies the Segal +condition in the first 푗 coordinates. +Proof. Claim (1): We need to check that (L푗푋)푚1,...,푚푘,0,•,...,• is a constant simplicial space. For푘 < 푗 −1 +this is true because constant simplicial spaces are Segal and hence Segalifying in the 푗th coordinate +does not change 푋푚1,...,푚푘,0,•,...,•. For 푘 = 푗 − 1 this is true because Segalifying never changes the +0-simplices. For 푘 ≥ 푗 consider the simplicial object 푌 : 횫op → Psh(횫×{푘+2,...,푛}) defined by sending +푙 to 푋푚1,...,푚푗−1,푙,푚푗+1,...,푚푘,0,•,···•. By assumption 푌푙 is a constant (푛 − 푘 − 1)-fold simplicial space for all +푙. Since the full subcategory of those (푛 − 푘 − 1)-fold simplicial spaces that are constant is closed +under all colimits, it follows that (Lseg푌)푙 is still constant for all 푙. +Claim (2): To simplify notation, we will assume that 푛 = 2 = 푗; the general case is analogous. +Suppose that 푋•,• satisfies the Segal condition in the first coordinate. It suffices to show that L2푋•,• +still satisfies the Segal condition in the first coordinate. In other words, we need to show that +(L2푋)•,푙 is a Segal space for all 푙. By the Segal condition in the second coordinate it suffices to do so +for 푙 = 0, 1. For 푙 = 0 there is nothing to show since Segalification does not change the 0-simplices. +For 푙 = 1 we have the necklace formula +(L2푋)•,1 ≃ +colim +Δ푛1∨···∨Δ푛푘 ∈Necop 푋•,푛1 ×푋•,0 · · · ×푋•,0 푋•,푛푘 +where the pullbacks and colimit are computed in simplicial spaces. +To complete the proof it +suffices to show that the diagram on the right hand side whose colimit we are taking satisfies the +hypotheses of Lemma 2.11. The indexing category Necop is sifted by Lemma 1.28 and each of the +terms in the diagram is a Segal space since Segal spaces are closed under pullbacks. It remains to +16 + +observe that the diagram of 0-simplices is constant since its value on any necklace 푁 = Δ푛1 ∨· · ·∨Δ푛푘 +is +푋0,푛1 ×푋0,0 · · · ×푋0,0 푋0,푛푘 ≃ 푋0,0 ×푋0,0 · · · ×푋0,0 푋0,0 ≃ 푋0,0 +by the reduced assumption. +□ +Proposition 2.13. The left adjoint to the full inclusion of 푛-fold Segal space into reduced 푛-fold simplicial +spaces may be computed as +L = L푛 ◦ · · · ◦ L1 : Seg푛−fold +횫op +⇄ Psh푟 (횫×푛) :inc +Proof. For any 푛-fold simplicial space the map 푌•,...,• → (L푗푌)•,...,• is local with respect to those +푛-fold simplicial spaces that are Segal in the 푗th coordinate. Therefore, for any 푛-fold simplicial +space 푋•,...,• all of the maps +푋•,...,• → (L1푋)•,...,• → (L2 ◦ L1)(푋)•,...,• → · · · → (L푛 ◦ · · · ◦ L1)(푋)•,...,• +are local with respect to the full subcategory of 푛-tuple Segal spaces. If we assume that 푋•,...,• is also +reduced, then it follows by inductively applying Lemma 2.12 that (L푗 ◦ · · · ◦ L1)(푋)•,...,• is reduced +and satisfies the Segal condition in the first 푗 coordinates. We have therefore shown that the map +푋•,...,• → (L푛 ◦ · · · ◦ L1)(푋)•,...,• +is local with respect to 푛-fold Segal spaces and that its target is an 푛-fold Segal space. Consequently +it exhibits the target as the localization onto the full subcategory Segn−fold +횫op +. +□ +Warning 2.14. In the context of Proposition 2.13 the order in which the Segalification functors are +applied is crucial. It is important to Segalify the 1-morphisms first, then the 2-morphisms, and so +on. If we were to apply L2 first and then L1 it would no longer clear that the result is L2-local as L1 +can break the Segal condition in the second simplicial direction. +2.3 +The Boardman-Vogt tensor product +In this section we show how to use the Segalification formula to give a new construction of the +Boardman-Vogt tensor product of ∞-operads. We begin with a brief recollection on the tensor +product of commutative monoids. +Recollection on tensor product of commutative monoids. For an ∞-category with products C we +let CMon(C) denote the ∞-category of commutative monoids in C, see [GGN16, §1]. In the case +C = S we simply write CMon := CMon(S). We let Cat⊗ +∞ := CMon(Cat∞) denote the ∞-category of +symmetric monoidal ∞-categories. By applying CMon(−) to the adjunction +C(−) ⊣ N• and using +that CMon(Psh(횫)) ≃ Fun(횫op, CMon) we obtain an adjunction: +C(−) : Fun(횫op, CMon) ⇄ Cat⊗ +∞ :N•. +The right adjoint here is the symmetric monoidal nerve, which in the 푛th level is given by N푛(D) = +Fun([푛], D)≃ with the pointwise symmetric monoidal structure. +17 + +Let C be a cartesian closed presentable ∞-category. Gepner, Groth and Nikolaus [GGN16] show +that CMon(C) admits a canonical symmetric monoidal structure ⊗ such that the free functor +F: C → CMon(C) is symmetric monoidal. Moreover, if C and D are presentable cartesian closed +and 퐿: C → D is a symmetric monoidal (i.e. finite product preserving) left adjoint they show the +induced functor 퐿: CMon(C) → CMon(D) is canonically symmetric monoidal [GGN16, Lemma +6.3.(ii)]. Segalification, completion, and +C(−) are examples of such 퐿. We record this for future use. +Corollary 2.15. All of the functors in the following commutative diagram are canonically symmetric +monoidal for the respective tensor product: +Fun(횫op, CMon) +Seg횫op(CMon) +CSeg횫op(CMon) +Cat⊗ +∞ +L퐶 +Lseg +C(−) +≃ +N• (−) +Some equifibered theory. A morphism of commutative monoids 푓 : 푀 → 푁 is said to be +equifibered if the canonical square +푀 × 푀 +푀 +푁 × 푁 +푁 +푓 ×푓 ++ ++ +푓 +is cartesian [BS22, Definition 2.1.4]. This notion was introduced in op. cit. for the purpose of +developing the theory of ∞-properads. A quintessential feature of equifibered maps is that a +morphism of free monoids 푓 : +F(푋) → +F(푌) is equifibered if and only if it is free, i.e. 푓 ≃ +F(푔) for +some map of spaces푔: 푋 → 푌. Equifibered maps can be thought of as a well-behaved generalization +of free maps, for example they form the right part of a factorization system on CMon. Further +details can be found in [BS22, §2]. +Observation 2.16. Equifibered maps between free monoids are closed under tensor product. +Indeed, the free functor +F: S → CMon is symmetric monoidal and by [BS22, Remark 2.1.7] induces +an equivalence onto the subcategory CMonfree,eqf ⊆ CMon of free monoids and equifibered maps. +Definition 2.17. A simplicial commutative monoid 푀 : 횫op → CMon is called ⊗-disjunctive if it +is right-CMoneqf-fibered where CMoneqf ⊆ CMon denotes the subcategory of equifibered mor- +phisms. +Remark 2.18. By [BS22, Lemma 3.2.15] the nerve N•C of a symmetric monoidal ∞-category C ∈ +Cat⊗ +∞ is ⊗-disjunctive, if and only if C is ⊗-disjunctive in the sense of [BS22, Definition 3.2.14]. That +is, if and only if for all 푥,푦 ∈ C the functor +⊗: C/푥 × C/푦 −→ C/푥 ⊗푦 +is an equivalence. +Observation 2.19. Let 푀 : 횫op → CMon be ⊗-disjunctive. Then 푀 is level-wise free if and only if +푀0 is free. Indeed, evaluation at the last vertex 푑0 ◦ · · · ◦ 푑0 : 푀푛 → 푀0 is equifibered so if 푀0 is free +the same holds for 푀푛 [BS22, Corollary 2.1.11]. +As a consequence of the necklace formula we have the following: +18 + +Lemma 2.20. Let 푀 ∈ Fun(횫op, CMon) be ⊗-disjunctive. Then Lseg(푀) is ⊗-disjunctive. +Proof. It suffices to check the conditions of Proposition 2.3. The first condition was verified in +Example 1.25. The second condition follows from [BS22, Lemma 2.1.22]. +□ +We now give a description of ∞-operads using equifibered maps. +Definition 2.21. A pre-operad is a Segal commutative monoid 푀 ∈ Seg횫op (CMon) which is ⊗- +disjunctive and level-wise free. +A pre-operad is called complete if its underlying Segal space +is. +Warning 2.22. Pre-operads in the sense of Definition 2.21 should not be confused with ∞- +preoperads in the sense of Lurie [Lur, §2.1.4]. Instead, in the language of [BS22], a pre-operad +is precisely a monic pre-properad. +Pre-operads are to ∞-operads what Segal spaces are to ∞-categories. Indeed the envelope functor +induces an equivalence between Lurie’s ∞-operads and complete pre-operads. +Theorem 2.23. Lurie’s monoidal envelope functor Env(−) : Op∞ → Cat⊗ +∞ is faithful (i.e. it induces a +monomorphism on mapping spaces). Moreover, the composite +Op∞ +Env(−) +−−−−−→ Cat⊗ +∞ +N•≃ CSeg(CMon) +identifies Op∞ with the (non-full) subcategory of CSeg(CMon) whose objects are complete pre-operads and +whose morphisms are equifibered natural transformations. +The first instance of this theorem can be found in the work of Haugseng–Kock [HK21], who showed +that the sliced functor Env: Op∞ → Cat⊗ +∞/Fin is fully faithful and characterised its image. Barkan– +Haugseng–Steinebrunner [BHS22] then gave an alternative characterization of the image, closely +related to pre-operads. The above formulation was given in [BS22, Theorem 3.2.13]. +Tensor product of ∞-operads. We can now apply the Necklace formula for Segalification to show +that pre-operads are closed under tensor product. +Proposition 2.24. The replete subcategory pOp∞ ⊆ Seg횫op (CMon) is closed under tensor product. +Proof. First we claim that if 푀 : 횫op → CMon is level-wise free and ⊗-disjunctive then the same +holds for Lseg푀. Indeed Lemma 2.20 shows that Lseg(푀) is ⊗-disjunctive and since Lseg(푀)0 ≃ 푀0 +is free the claim follows from Observation 2.19. +To complete the proof it suffices to show that if 푀, 푁 ∈ Fun(횫op, CMon) are ⊗-disjunctive and +level-wise free, then the same holds for their tensor product 푀 ⊗ 푁. This follows from the fact that +equifibered maps between free monoids are closed under tensor product (see Observation 2.16). +□ +We are now in a position to prove Theorem E. +19 + +Proof of Theorem E. For the first part it suffices by Theorem 2.23 to show that complete pre-operads +are closed under the tensor product. By Corollary 2.15 the equivalence Cat⊗ +∞ ≃ CSeg횫op (CMon) +identifies the tensor product of symmetric monoidal ∞-categories with the following bi-functor on +complete Segal monoids +(푀•, 푁•) ↦−→ LCSS(푀• ⊗ 푁•) ≃ L퐶Lseg(푀• ⊗ 푁•). +Suppose now that 푀• and 푁• are pre-operads. By Proposition 2.24, Lseg(푀• ⊗ 푁•) is a pre-operad +and hence [BS22, Proposition 3.4.7] so is the completion L퐶Lseg(푀• ⊗ 푁•). +For the second part we must compare ⊗BV to Lurie’s tensor product. It follows from Lemma 2.25 +that there is an equivalence Env(O ⊗Lurie P) ≃ Env(O ⊗BV P) for all ∞-operads O and P. And +since Env is an equivalence onto a replete subcategory by Theorem 2.23 it follows that we already +have such an equivalence before applying Env. +□ +Lemma 2.25. Lurie’s tensor product satisfies that for any two ∞-operads O and P there is a canonical +equivalence: +Env(O ⊗Lurie P) ≃ Env(O) ⊗ Env(P) +Proof. The tensor product in Cat⊗ +∞ is adjoint to the internal hom: +FunCat⊗ +∞ (C ⊗ D, E ) ≃ FunCat⊗ +∞(C, FunCat⊗ +∞ (D, E )), +and monoidal envelope Env: Op∞ → Cat⊗ +∞ is left adjoint to the forgetful functor Cat⊗ +∞ → Op∞ and +by [Lur, Proposition 2.2.4.9.] we have an equivalence: +FunCat⊗ +∞ (Env(O), E ) ≃ AlgO(E ). +Combining these facts with the key property5 of Lurie’s tensor product that AlgO⊗LurieP (E ) ≃ +AlgO(AlgP (E )) we obtain a sequence of equivalences: +FunCat⊗ +∞(Env(O ⊗Lurie P), E ) ≃ AlgO⊗LurieP (E ) ≃ AlgO(AlgP (E )) +≃ FunCat⊗ +∞ (Env(O), FunCat⊗ +∞(Env(P), E )) +≃ FunCat⊗ +∞ (Env(O) ⊗ Env(P), E ). +As this is natural in E ∈ Cat⊗ +∞ the claim follows from the Yoneda lemma. +□ +References +[Bar] +Clark Barwick. (∞,푛)-Cat as a closed model category. 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In: North- +Western European Journal of Mathematics 3 (2017), pp. 141–202. +21 + diff --git a/_tFAT4oBgHgl3EQfrB0d/content/tmp_files/load_file.txt b/_tFAT4oBgHgl3EQfrB0d/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..64588873eec44b5bd4806e9eb1df4d00a90cc17d --- /dev/null +++ b/_tFAT4oBgHgl3EQfrB0d/content/tmp_files/load_file.txt @@ -0,0 +1,1137 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf,len=1136 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='08650v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='AT] 20 Jan 2023 Segalification and the Boardman–Vogt tensor product Shaul Barkan and Jan Steinebrunner January 23, 2023 Abstract We develop an analog of Dugger and Spivak’s necklace formula providing an explicit de- scription of the Segal space generated by an arbitrary simplicial space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We apply this to obtain a formula for the Segalification of 푛-fold simplicial spaces, a new proof of the invariance of right fibrations, and a new construction of the Boardman–Vogt tensor product of ∞-operads for, which we also derive an explicit formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Contents 1 Segalification 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='1 Necklace contexts .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 9 2 Applications 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='1 Segalification and right fibrations .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='2 Segalification for (∞,푛)-categories .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='3 The Boardman-Vogt tensor product .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 17 References 20 Introduction A particularly useful and robust perspective on ∞-categories is provided by simplicial spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Recall that a Segal space is a simplicial space 푋 : 횫op → S for which the natural map 푋푛 ≃−→ 푋1 ×푋0 · · ×푋0 푋1 is an equivalence for all 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' A Segal space is complete if the map 푠0 : 푋0 → 푋1 induces an equivalence onto a certain union of components 푋 eq 1 ⊂ 푋1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We let PshCSS(횫) ⊂ Pshseg(횫) ⊂ Psh(횫) denote the full subcategories of (complete) Segal spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' There are localizations: PshCSS(횫) Pshseg(횫) Psh(횫) L퐶 Lseg ⊣ ⊣ 1 and we denote the composite left adjoint by LCSS : Psh(횫) → PshCSS(횫).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The ∞-category Cat∞ of ∞-categories participates in an adjunction C: Psh(횫) ⇄ Cat∞ :N• where the left adjoint C is left Kan extended from the inclusion 횫 ⊂ Cat∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The right adjoint N• is given by restricting the Yoneda embedding along 횫 ⊂ Cat∞ and induces an equivalence N• : Cat∞ ≃ PshCSS(횫) under which C(−) is identified with LCSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' To avoid confusion we will write LCSS for the localization endofunctor of Psh(횫) and C(−) for the left adjoint with values in Cat∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' In his foundational work on complete Segal spaces [Rez01], Rezk provides a formula for the Rezk- completion functor L퐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The purpose of this note is to provide a formula for the “Segalification” functor Lseg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Combining the two one obtains an explicit description of the ∞-category generated by an arbitrary simplicical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Necklaces and Segalification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Our formula for Lseg is heavily influenced by the work of Dugger and Spivak [DS11] on the rigidification of quasicategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' It will involve a colimit indexed by a certain category of “necklaces” [DS11, §3] which we now recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' A necklace is a simplicial set 푁 = Δ푛1 ∨ · · · ∨ Δ푛푘 obtained by joining standard simplices at their start and endpoints, as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Following Dugger and Spivak we define Nec to be the (non-full) subcategory of sSet 휆 Figure 1: A morphism of necklaces 휆: Δ2 ∨ Δ2 ∨ Δ1 ∨ Δ1 → Δ1 ∨ Δ3 ∨ Δ2 defined by collapsing the first edge and including the remaining necklace as indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' whose objects are necklaces and whose morphisms are maps of simplicial sets 푓 : Δ푛1 ∨ · · · ∨ Δ푛푘 → Δ푚1 ∨ · · · ∨ Δ푚푙 that preserve the minimal and maximal elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We can now state the formula for Lseg in terms of necklaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' For the sake of simplicity we state here the formula only for the case of 1-simplices Lseg(푋)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' This suffices to determine L(푋)푛 for all 푛 by the Segal condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' For every simplicial space 푋 ∈ Psh(횫) there is a canonical equivalence: LSeg(푋)1 ≃ colim 푁 ∈Necop MapPsh(횫) (푁, 푋) ≃ colim Δ푛1∨···∨Δ푛푟 ∈Necop 푋푛1 ×푋0 · · · ×푋0 푋푛푟 One could in fact deduce this theorem from the results of Dugger–Spivak by passing through the various model categories for ∞-categories, but we will instead give a “synthetic” proof as we believe it to be insightful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Application: right fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Our first application is to the notion of right fibrations of simplicial spaces in the sense of Rezk (see [Ras17, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Using the Segalification formula we show that right fibrations of simplicial spaces are invariant under LCSS-equivalences, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' if 푓 : 푋 → 푌 is 2 a map of simplicial spaces such that LCSS(푓 ) is an equivalence then base change along 푓 induces an equivalence 푓 ∗ : Psh(횫)r−fib /푌 ≃ Psh(횫)r−fib /푌 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' This implies that right fibrations over an arbitrary simplicial space 푋 model presheaves on the associated ∞-category C(푋): Corollary B (Rasekh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' For any simplicial space 푋 the functor C(−) induces an equivalence Psh(Δ)r−fib /푋 C(−) −−−−→ ≃ Catr−fib ∞/C(푋) ≃ Psh( C(푋)) A model-categorical version of this result was previously proven by Rasekh [Ras17, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='18 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Our proof has the advantage of being “synthetic” and also significantly shorter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' An alternative formulation of this corollary is to say that (the nerve of) the universal right fibration Sop ∗ → Sop classifies right fibrations of arbitrary simplicial spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='1 We shall now use the above result to give a formula for C(−) : Psh(횫) → Cat∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Let us write 횫max ⊆ 횫 for the wide subcategory spanned by morphisms which preserve the maximal element, and given a simplicial space 푋 : 횫op → S let us write 푝푋 : 횫/푋 → 횫 for the associated right fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' For every simplicial space 푋 ∈ Psh(횫), the last vertex functor 횫/푋 → C(푋) induces an equivalence of ∞-categories 횫/푋 [푊 −1 푋 ] ≃ C(푋) , where 푊푋 ≔ 푝−1 푋 (횫max) ⊆ 횫/푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' In the case that 푋 is level-wise discrete, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' a simplicial set, this recovers a result of Stevenson [Ste17, Theorem 3] who attributes it to Joyal [Joy07, §13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Application: (∞,푛)-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The ∞-category of (∞,푛)-categories admits many equivalent de- scriptions including Rezk’s complete Segal Θ푛-spaces [Rez10] and Barwick’s complete 푛-fold Segal spaces [Bar].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' These were shown to be equivalent by Barwick–Schommer-Pries [BS21] and later us- ing different techniques also by Bergner–Rezk [BR20] and Haugseng [Hau18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We refer the reader to [Hau21, §2] for a brief account of different models and some known comparisons between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We shall now present an application of our main result to 푛-fold Segal spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We say that an 푛-fold simplicial space 푋 : (횫op)×푛 → S is reduced if each of the (푛 − 푘 − 1)-fold simplicial spaces 푋푚1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',푚푘,0,•,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',• is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We write Psh푟 (횫×푛) ⊆ Psh(횫×푛) for the full subcategory of reduced 푛-fold simplicial spaces and let Seg푛−fold 횫op ⊆ Psh푟 (횫×푛) denote the full subcategory spanned by the 푛-fold Segal spaces, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' those that satisfy the Segal condition in each coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' This inclusion Seg푛−fold 횫op ↩→ Psh푟 (횫×푛) admits a left adjoint and we give a formula for it: Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The left adjoint L: Psh푟 (횫×푛) → Seg푛−fold 횫op may be computed as L = L푛 ◦ · · · ◦ L1 where L푗 : Psh(횫×푛) → Psh(횫×푛) denotes the endofunctor that segalifies the 푗th coordinate: (L푗푋)푚1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',푚푗−1,1,푚푗+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',푚푛 ≃ colim Δ푛1∨···∨Δ푛푟 ∈Necop 푋푚1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',푛1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',푚푛 × 푋푚1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',푚푛 · · × 푋푚1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',푚푛 푋푚1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',푛푟,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',푚푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Note that here the order of the L푗 is crucial: we need to first Segalify 1-morphisms, then 2- morphisms, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' If one were to apply L2 and then L1 the result would not necessarily satisfy the Segal condition in the second coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 1Note that the content of this statement is not so much that it identifies the universal right fibration of simplicial spaces (this can be done easily since the simplices Δ푛 are complete Segal spaces), but rather that it states that right fibrations of simplicial spaces admit a universal example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 3 Application: the Boardmann–Vogt tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' One of the many great achievements of Lurie’s book project on Higher Algebra [Lur] is the construction of a homotopy coherent symmetric monoidal structure ⊗Lurie on the ∞-category of ∞-operads Op∞ generalizing the Boardmann-Vogt tensor product [BV73, §II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The defining property of O ⊗Lurie P is that algebras over it are “O- algebras in P-algebras”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' that for any symmetric monoidal ∞-category C there is an equivalence AlgO⊗LurieP (C) ≃ AlgO(AlgP (C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The construction of ⊗Lurie is quite intricate as it involves a delicate mix of quasicategorical and model categorical techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We shall now describe how the necklace formula can be used to justify a simpler alternative approach to the tensor product of ∞-operads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The tensor product of symmetric monoidal ∞-categories uniquely restricts to a tensor product ⊗BV on Op∞ such that the envelope Env: (Op∞, ⊗BV) → (Cat⊗ ∞, ⊗) is a symmetric monoidal functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Moreover, for any two ∞-operads O and P there is a canonical equivalence O ⊗BV P ≃ O ⊗Lurie P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Note that Theorem E does not claim that (Op∞, ⊗BV) and (Op∞, ⊗Lurie) are equivalent as symmetric monoidal ∞-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' It does however reduce the question to whether the envelope can be constructed as a symmetric monoidal functor (Op∞, ⊗Lurie) � (Cat⊗ ∞, ⊗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' This is not entirely clear since the higher coherence data (associator, braiding, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' ) of Lurie’s tensor product ⊗Lurie is tricky to access2 as it seemingly has no obvious universal property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Moreover, the authors are unaware of any applications in which the specific coherence of Lurie’s construction plays a role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' A novel consequence of Theorem E is that, at least in principle, the Boardman-Vogt tensor product is only as difficult to compute as necklace colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The resulting formula will be easiest to express in the language of symmetric sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Outlook: symmetric sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' A symmetric sequence is a presheaf on the category of finite sets and bijections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The disjoint union ⊔ and the product × of finite sets extend via Day convolu- tion to symmetric monoidal structures on symmetric sequences SymSeq ≔ Psh(Fin�) which we respectively denote by ⊗ and ⊠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Since (SymSeq, ⊗) is the free presentably symmetric monoidal ∞-category on a single generator 1 ∈ SymSeq, evaluation induces an equivalence ev1 : FunCAlg(PrL) �(SymSeq, ⊗), (SymSeq, ⊗)� ≃−→ SymSeq which endows SymSeq with yet another (non-symmetric) monoidal structure ◦ coming from the composition of endofunctors on the left side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' It is generally expected that 1-colored (non-complete) ∞-operads are equivalent to associative algebras for ◦ in SymSeq, and for a different definition of ◦ this was shown in [Hau22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Given two such algebras O, P ∈ AlgE1(SymSeq, ◦) the necklace formula in this setting gives: O ⊗BV P ≃ colim Δ푛1∨···∨Δ푛푟 ∈Necop (O◦푛1 ⊠ P◦푛1) ◦ · · · ◦ (O◦푛푟 ⊠ P◦푛푟 ) A formal proof of this does not fit within the scope of the present paper, as it relies on a good interface between ∞-operads and symmetric sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We plan to provide such an interface and provide a full proof of this in forthcoming work where we revisit the composition product of symmetric sequences from the perspective of equifibered theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 2Lurie does give a model categorical construction of a (non-symmetric) monoidal structure which does have a recog- nizable universal property as a certain localization of ∞-categories over Fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The symmetric monoidal structure however is constructed by hand and apart from the binary operation the relation between the two is not commented on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 4 Acknowledgments We would like to thank Rune Haugseng for helpful comments on an earlier draft and Maxime Ramzi for useful conversations related to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The first author would like to thank the Hausdorff Research Institute for Mathematics for their hospitality during the fall 2022 trimester program, funded by the Deutsche Forschungsgemein- schaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2047/1 – 390685813.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The second author is supported by the Danish National Research Foundation through the Copenhagen Centre for Geometry and Topology (DNRF151).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 1 Segalification 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='1 Necklace contexts Let us fix an ∞-category C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' In this section we study the general problem of approximating a reflective localization functor L: Psh(C) → Psh(C) in the sense of [Lur09, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='4] using a suitable auxiliary subcategory of Psh(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' A presheaf 푋 ∈ Psh(C) is called L-local if the canonical map 푋 → L(푋) is an equivalence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' if 푋 lies in the essential image of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' A morphism of presheaves 푓 : 푌 → 푍 ∈ Psh(C) is called an L-local equivalence if L(푓 ) is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' A necklace context is a triple (C, L, N ) where C and L are as above and N ⊆ Psh(C) is a full subcategory such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Y(푐) ≔ MapC (−,푐) is L-local for all 푐 ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Y(푐) ∈ N for all 푐 ∈ C and L(푁) ∈ Y(C) for all 푁 ∈ N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' A compatible necklace category for a pair (C, L) as in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='2 exists if and only if the first condition holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' In this case, the minimal possible necklace category is given by the representable presheaves Nmin ≔ Y(C) ⊆ Psh(C) and the maximal choice is given by Nmax ≔ L−1(C) ⊆ Psh(C), namely all 푋 ∈ Psh(C) such that L(푋) ∈ Y(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The full subcategory Nsub ⊆ Psh(C) spanned by all subobjects 퐴 ⊆ Y(푐) such that L(퐴) ≃ Y(푐) is another possible choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Given a necklace context (C, L, N ), the Yoneda embedding Y : C ↩→ Psh(C) lands in N ⊆ Psh(C) and thus gives rise to an adjunction 푙 ≔ L|N : N C :Y =:푖 ⊣ Passing to to presheaves we obtain a quadruple adjunction Psh(C) Psh(N ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 푖!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='=푙∗ 푖∗=푙∗ 푙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 푖∗ ⊣ ⊣ ⊣ 5 Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The natural transformation 훽 : 푖∗ → 푙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' defined by � 훽 : 푖∗ 푖∗◦푢 −−−→ 푖∗ ◦ (푙∗ ◦ 푙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=') ≃ (푙 ◦ 푖)∗ ◦ 푙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' ≃ 푙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' � ∈ Fun(Psh(N ), Psh(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' is L-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The functors L ◦ 푖∗ and L ◦ 푙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' are both left adjoints, when thought of as functors Psh(N ) −→ PshL−loc(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Therefore it will suffice to check that L(훽푍) is an equivalence for representable presheaves 푍 = Y푁 with 푁 ∈ N , as these generate Psh(N ) under colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Restricting Y푁 along 푖 we obtain 푖∗Y푁 = 푁 ∈ Psh(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' One the other hand we have 푙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='Y푁 ≃ Y푙 (푁) ≃ L(푁) and the natural transformation in this case is the canonical map 훽Y푁 : 푁 → L(푁) which is indeed L-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' □ Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Given a necklace category (C, L, N ) we define 푄N : Psh(C) 푖∗−→ Psh(N ) 푙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='−→ Psh(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' This functor receives a canonical natural transformation from the identity 휆 : IdPsh(C) ≃ ←− 푖∗ ◦ 푖∗ 훽◦푖∗ −−−→ 푙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' ◦ 푖∗ = 푄N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The functor 푖∗ : Psh(C) → Psh(N ) may be computed as (푖∗푋)(푁) ≃ MapPsh(C) (푁,푋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' By the pointwise formula for left Kan extensions thus have 푄N (푋)(푐) = colim (푁,푙 (푁)←푐) ∈(N ×C C푐/)op MapPsh(C) (푁, 푋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Note that by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='4, 휆: id → 푄N (푋) is L-local and thus the unit transformation id → L factors through 휆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The resulting natural transformation 푄N → L is then L-local by cancellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We thus conclude: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' There exists a canonical L-local natural transformation 푄N → L such that for any 푋 ∈ Psh(C) the map 푄N (푋) → L(푋) is an equivalence if and only if 푄N (푋) is L-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Since 푄N is accessible and 휆: id → 푄N satisfies QN ◦ 휆 ≃ 휆QN : QN → Q2 N , we can iterate 푄N transfinitely (in analogy with how sheafification is constructed in [Lur09, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='7]) to obtain a localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' This localization will be equivalent to L if and only the morphisms {푁 → L(푁)}푁 ∈N generate the class of L-local equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='2 Segalification We now specialize to the setting of Segal spaces, where the localization Lseg is defined as the left adjoint to the full inclusion Seg횫op (S) ↩→ Fun(횫op, S) of those simplicial spaces 푋• for which the map 푋푛 → 푋1 ×푋0 · · · ×푋0 푋1 is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We will choose a necklace category and show that 푄N ≃ Lseg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 6 Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The formula we will arrive at for Lseg is closely related to the work of Dugger–Spivak [DS11], who construct a functor ℭnec : sSet → sCat from the category of simplicial sets to the category of simplicial categories, which they show to be weakly equivalent to the left adjoint ℭ of the coherent nerve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' This gives a formula for the mapping spaces in ℭ(푍•) (as a colimit over the necklace category) when 푍• is a simplicial set in the Joyal model structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The case of a simplicial space follows by using the left Quillen equivalence 푡!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' : ssSet → sSet constructed by Joyal–Tierney [JT06].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Segal spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Let us recall the category of necklaces, introduced by Dugger and Spivak [DS11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The concatenation 퐴 ∨ 퐵 of two bi-pointed simplicial sets (퐴,푎min,푎max) and (퐵,푏min,푏max) is defined as the pushout Δ0 퐵 퐴 퐴 ∨ 퐵, 푎max 푏min ⌟ which we point as (퐴 ∨ 퐵,푎min,푏max).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' This defines a (non-symmetric) monoidal structure on the category of bi-pointed simplicial sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' A necklace is a bi-pointed simplicial set obtained by concatenating simplices (Δ푛, 0,푛), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' it is of the form 푁 = Δ푛1 ∨ · · · ∨ Δ푛푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We let Nec denote the category whose objects are necklaces and whose morphisms are maps of bi-pointed simplicial sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' While the category Nec will play the central role in the Segalification formula, we will need a slightly bigger category to set up the necklace context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Let N ⊂ Psh(횫) denote the essential image of the faithful functor Nec → Psh(횫).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Segalification Lseg : Psh(횫) → Seg횫op (S) restricts to a functor Lseg|N : N → 횫.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' In particular the triple (횫, Lseg, N ) is a necklace context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We will show by induction on 푛 that the inclusion Δ{0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',푛1 } ∨ · · · ∨ Δ{푛푘,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',푛} ↩→ Δ푛 is a Segal equivalence thereby proving the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Consider the nested inclusion Δ{0,1} ∨ · · · ∨ Δ{푛−1,푛} ↩→ Δ{0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',푛1 } ∨ · · · ∨ Δ{푛푘,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',푛} ↩→ Δ푛 The composite is a Segal equivalence by definition and since L preserves colimits and 푛푗+1 − 푛푗 < 푛 the first map is a Segal equivalence by the induction hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The second map is therefore a Segal equivalence by cancellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' □ Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Note that the map 푁 → Lseg(푁) = Δ푛 is a monomorphism for each necklace 푁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' In particular, for any two necklaces 푁, 푀, the map MapN (푁, 푀) −→ MapPsh(횫) (Lseg(푁), Lseg(푀)) ≃ Map횫([푛], [푚]) is a monomorphism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' L: N → 횫 is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 7 The Segal condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Since (횫, Lseg, N ) is a necklace context we have by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='13 a functor 푄 : Psh(횫) 푖∗−→ Psh(N ) 푙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='−→ Psh(횫).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' By Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='7 this comes with a Lseg-local natural transformation 푄 → Lseg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We may compute the functor 푄(−) using Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='6 as: 푄(푋)푛 = colim (푁,푙 (푁)←[푛]) ∈(N ×횫횫[푛]/)op MapPsh(횫) (푁, 푋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Below we show that 푄(푋) is always a Segal space and thus by Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='7 that 푄 ≃ Lseg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' For any necklace 푁 we let 휄푁 : Δ1 → Lseg(푁) denote the unique map that preserves the extrema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Given [푛] ∈ 횫 we define the functor 퐽 : 푛 � 푖=1 Nec ↩→ N ×횫 횫[푛]/, by joining necklaces at their endpoints (푀1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' , 푀푛) ↦−→ � 푀1 ∨ · · · ∨ 푀푛, [푛] Lseg (휄1∨···∨휄푛) −−−−−−−−−−−→ Lseg(Lseg(푀1) ∨ · · · ∨ Lseg(푀푛)) ≃ Lseg(푀1 ∨ · · · ∨ 푀푛) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The functor 퐽 is fully faithful and admits a right adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' In particular it is coinitial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Fully-faithfulness follows by unwinding definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We claim that a right adjoint to 퐽 is given by the following 퐽 푅 : (푁, 훼 : [푛] → Lseg(푁)) ↦−→ (푁훼 (0),훼 (1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' , 푁훼 (푛−1),훼 (푛)) where 푁훼 (푗),훼 (푗+1) ≔ 푁 ∩ Δ{훼 (푗),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',훼 (푗+1) }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' To see this, note that a tuple of necklace morphisms (푀1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' , 푀푛) → (푁훼 (0),훼 (1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' , 푁훼 (푛−1),훼 (푛)) = 퐽 푅(푁, 훼) is equivalent to a morphism 푀1 ∨ · · · ∨ 푀푛 → 푁훼 (0),훼 (1) ∨ · · · ∨ 푁훼 (푛−1),훼 (푛) ⊂ 푁 such that 푀푖 lands in 푁훼 (푖−1),훼 (푖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' These can be identified with morphisms 퐽 (푀1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' , 푀푛) → (푁,훼) in N ×횫 횫[푛]/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Therefore 퐽 푅 is indeed right adjoint to 퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' □ Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The cofinality in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='16 implies that 푄(푋)푛 may be computed as: 푄(푋)푛 ≃ colim (푀1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',푀푛) ∈(Necop)푛 MapPsh(횫) (푀1 ∨ · · · ∨ 푀푛,푋) ≃ colim (푀1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',푀푛) ∈(Necop)푛 MapPsh(횫) (푀1, 푋) × 푋0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' × 푋0 MapPsh(횫) (푀푛, 푋) In particular for 푛 = 0 we just get 푄(푋)0 = 푋0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' While this is a simplification of the general formula from Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='6, it has the downside that the functoriality in [푛] is not clear in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' However, we can still see the functoriality in inert maps 휑 : [푚] \u058c [푛] as it is simply given by restricting to the 푀푖 that correspond to the image of 휑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' This functoriality will suffice to check the Segal condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' For any simplicial space 푋• the simplicial space 푄(푋)• is a Segal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 8 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Consider the following diagram: colim (푀1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',푀푛) ∈(Necop)푛 Map(푀1,푋) × 푋0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' × 푋0 Map(푀푛,푋) 푄(푋)푛 colim 푀1 ∈Necop Map(푀1, 푋) × 푋0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' × 푋0 colim 푀푛 ∈Necop Map(푀1,푋) 푄(푋)1 × 푄 (푋)0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' × 푄 (푋)0 푄(푋)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The horizontal maps are equivalences by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='16 and Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The left vertical map is an equivalence since the cartesian product in S/푋0 preserves colimits in each variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' □ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='3 Variations on the Segalification formula A formula for mapping spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Given a Segal space 푋 ∈ Seg횫op(S) the mapping spaces in the associated ∞-category C(푋) may be computed as Map C(푋) (푥,푦) ≃ {푥} ×푋0 푋1 ×푋0 {푦} for any 푥,푦 ∈ 푋0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Below we show how to use the results of the previous to derive a formula for these mapping spaces when 푋 is an arbitrary simplicial space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' In the case where 푋 is a simplicial set this recovers the formula of Dugger–Spivak [DS11], which inspired our Segalification formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' For any simplicial space 푋 ∈ Psh(횫) and 푥,푦 ∈ 푋 there is a canonical equivalence |Nec/(푋,푥,푦) | ≃ Map C(푋) (푥,푦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' where Nec/(푋,푥,푦) ⊆ � Psh(횫)Δ0⊔Δ0/ � /(푋,푥,푦) denotes the full subcategory spanned by necklaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Interpreting Nec as a full subcategory of Psh(횫)Δ0⊔Δ0/ by recording the minimal and maximal vertex we can fit Nec/(푋,푥,푦) into a cartesian square: Nec/(푋,푥,푦) Nec ×Psh(횫) Psh(횫)/푋 {(푥,푦)} 푋0 × 푋0 The top right corner is a right fibration over Nec corresponding to the presheaf 푖∗(푋) : Necop → S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Its weak homotopy type is thus the colimit of this functor, which is precisely the definition of 푄(푋)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The bottom edge in the square is a map of spaces, hence inverting all maps we obtain |Nec/(푋,푥,푦) | ≃ {(푥,푦)} ×푋 ×2 0 푄(푋)1 ≃ Map C(푋) (푥,푦), where the second equivalence holds since the Rezk-completion of 푄(푋) ≃ Lseg(푋) is the nerve of C(푋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' □ Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Given three points 푥,푦, 푧 ∈ 푋 the monoidal structure ∨ on Nec yields a functor ∨: Nec/(푋,푥,푦) × Nec/(푋,푦,푧) −→ Nec/(푋,푥,푧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Onweakhomotopytypesthisyieldsthecomposition Map C(푋) (푥,푦)×Map C(푋) (푦,푧) → Map C(푋) (푥,푧) in C(푋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' (As in [DS11, Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=') This can be seen by an argument similar to Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='19 to the necklace formula for 푄2푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 9 1-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Similar to how Δop-colimits in 1-categories can be computed as reflexive coequalizers, Necop colimits in a 1-category can be reduced to certain “thin” necklaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We say that a necklace 푁 = Δ푛1 ∨ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Δ푛푟 ∈ Nec is thin if � 푖 푛푖 ≤ 푟 + 1 and 푛푖 ≥ 1, in other words if it consists of 1-simplices and at most one two-simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' If 푁 consists only of 1-simplices we say that it is very thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Let Necthin ⊂ Nec denote the full subcategory of thin necklaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The full inclusion Necop thin ↩→ Necop is 1-cofinal, that is, for any functor Necop → C to a 1-category C the colimit may equivalently be computed over Necop thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We need to show that for any necklace 푁 = Δ푛1 ∨ · · · ∨ Δ푛푟 ∈ Nec the slice category Necthin/푁 is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We enumerate the vertices of 푁 in their canonical order as 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' ,푛 = � 푖 푛푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' A very thin necklace over 푁, Δ1 ∨ · · · ∨ Δ1 → 푁 may equivalently be encoded as a non-decreasing path 0 = 푎0 ≤ · · · ≤ 푎푘 = 푛 in [푛].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' These paths are subject to the condition that we never have strict inequalities 푎푙 < �푠 푖=1 푛푖 < 푎푙+1 for any푠 and푙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Suppose that 푝 = (0 = 푎0 ≤ · · · ≤ 푎푘 = 푛) is such a path and 푠 is such that 푝′ = (0 = 푎0 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' �푎푠 · · · ≤ 푎푘 = 푛) is still an admissible path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Then there is a thin necklace 푀 with a two-simplex (푎푠−1 ≤ 푎푠 ≤ 푎푠+1) that contains both of these paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' In particular, the paths are connected through a zig-zag 푝 → 푀 ← 푝′ as objects of Necthin/푁 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Proceeding be removing a vertex whenever possible we see that every very thin necklace over 푁 is connected in Necthin/푁 to a very thin necklace that corresponds to a minimal path in 푁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' But there is only one path in 푁 that is minimal with respect to removing vertices, namely (0 ≤ 푛1 ≤ · · · ≤ �푟−1 푖=1 푛푖 ≤ 푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Therefore all the very thin objects in Necthin/푁 are connected, and thus the category is connected as every (thin) necklace contains a very thin necklace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' □ Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Suppose that푋• is a simplicial space and we want to compute the homotopy category ℎ1( C(푋)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' For simplicity, let us assume that 푋푛 is discrete for all 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='3 Then the set of morphisms in ℎ1( C(푋))) is exactly 휋0(Lseg(푋)1) and we may compute it as the colimit Mor(ℎ1( C(푋))) � colim 푁 ∈Necop Map(푁, 푋) � colim Δ푛1∨···∨Δ푛푟 ∈Necop 푋 (Δ푛1) ×푋 (Δ0) · · · ×푋 (Δ0) 푋 (Δ푛푟 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' in the 1-category of sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' By Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='22 it suffices to take the colimit over Necop thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Moreover the very thin necklaces are 0-cofinal, so the colimit may be expressed as a coproduct over the very thin necklaces modulo an equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' This leads to the formula Mor(ℎ1( C(푋))) � �� 푛≥0 푋1 ×푋0 · · · ×푋0 푋1 � /∼ where the equivalence relation is generated by (푓1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' , 푓푛) ∼ (푓1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' , 푓푖−1,푔, 푓푖+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' , 푓푛) whenever there is a 2-simplex in 푋 witnessing 푓푖+1 ◦ 푓푖 = 푔, and (푓1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' , 푓푛) ∼ (푓1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' , 푓푖−1, 푓푖+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' , 푓푛) whenever 푓푖 is a degenerate 1-simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' This recovers the classical formula for the homotopy category of a simplicial set: namely, it is the free category on the edges of 푋 modulo the relations generated by the 2-simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 3This is not a very restrictive assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Starting with a general simplicial space 푌• we may base change it along a 휋0-surjective map 푍0 → 푌0 to get a simplicial space 푍푛 = (푍0)푛+1 ×푌푛+1 0 푌푛 such that the resulting functor C(푍•) → C(푌•) will be an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Now we may choose 푍0 to be discrete and define 푋푛 ≔ 휋0(푍푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Then C(푍•) → C(푋•) induces an equivalence on homotopy categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 10 Segalificationin other categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' In this section we establish criteria on a presentable ∞-category, which guarantee that Segalification is given by the necklace formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' This is summarized by the following result, which we prove in the remainder of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Let V be a presentable ∞-category in which sifted colimits are stable under base change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Then the left adjoint to the inclusion Seg횫op (V) ↩→ Fun(횫op, V) is given by the necklace formula: Lseg(푋)1 ≃ colim 푁 ∈Necop(푖∗푋)(푁) ≃ colim Δ푛1∨···∨Δ푛푘 ∈Necop 푋푛1 ×푋0 · · · ×푋0 푋푛푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' where 푖∗ denotes the right Kan extension 푖∗ : Fun(횫op, V) → Fun(N op, V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Recall that if 픛 is an ∞-topos then all colimits in 픛 are universal [Lur09, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' In particular Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='24 applies to ∞-topoi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' A wider variety of examples is provided by passing to algebras over ∞-operads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Let V be a presentably symmetric monoidal ∞-category4 and O be an ∞-operad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Then the forgetful functor AlgO(V) → V creates and preserves both limits and sifted colimits [Lur, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='4 & 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Consequently, if sifted colimits in V are stable under base change, then the same holds for AlgO(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='24 holds if we assume that Necop-colimits in V are stable under base change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The left adjoint Lseg exists by the adjoint functor theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Since V is presentable we may find a small ∞-category E and a fully faithful right adjoint 퐼 : V ↩→ Psh(E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We denote the resulting adjunction on presheaf categories by 퐼 횫 : Fun(횫op, V) ⇄ Fun(횫op, Psh(E )) :퐿횫 We now define an endofunctor 푄V ≔ 퐿횫 ◦ 푄 ◦ 퐼 횫 : Fun(횫op, V) → Fun(횫op, V) where 푄 is the endofunctor on Fun(횫op, Psh(E )) ≃ Fun(E , Psh(횫)) given pointwise given by the usual formula (see Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' This 푄V receives a natural transformation 휆V : Id ≃ 퐿횫 ◦ 퐼 횫 퐿횫◦휆◦퐼횫 −−−−−−→ 퐿횫 ◦ 푄 ◦ 퐼 횫 = 푄V coming from 휆: id → 푄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' This transformation is a Segal equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Indeed, if 푋,푌 : 횫op → V and 푌 is Segal, then in the commutative square MapFun(횫op,V)(푄V푋,푌) MapFun(횫op,V) ((퐿횫 ◦ 퐼 횫)(푋),푌) MapFun(횫op,V)((푄 ◦ 퐼 횫)(푋), 퐼 횫(푌)) MapFun(횫op,V) (퐼 횫(푋), 퐼 횫(푌)) ≃ ≃ (−)◦휆V (−)◦휆 the bottom map is an equivalence since 퐼 횫(푌) is Segal and thus so is the top map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' It remains to show that 푄V (푋) is Segal for all 푋 : 횫op → V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' This follows from the same proof as Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='18 by using that Necop-colimits are stable under base change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' □ 4In fact it suffices to ask that the monoidal structure is compatible with sifted colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 11 Diagrams of shape Necop may be difficult to recognize in the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Fortunately, Necop is a sifted category, colimits over which are well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We will deduce this from the following fact to which it is intimately linked: Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The Segalification functor L: Psh(Δ) → Psh(Δ) preserves products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The two functors Psh(Δ) × Psh(Δ) −→ Cat∞ given by (푋,푌) ↦→ L(푋 × 푌) and L(푋) × L(푌) both preserve colimits in both variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Therefore it suffices to check that the natural transformation between them is an equivalence on (Δ푛, Δ푚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' But in this case it is easy to check because Δ푛, Δ푚 and Δ푛 × Δ푚 are all Segal spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' □ Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The category Necop is sifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We need to show that the diagonal functor Δ: Necop → Necop×Necop is cofinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Equivalently, we need that for all 퐴, 퐵 ∈ Nec the slice Nec ×Nec2 Nec2 /(퐴,퐵) is weakly contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' This category is equivalent to the full subcategory Nec/퐴×퐵 ⊆ � Psh(횫)Δ0⊔Δ0/ � /퐴×퐵 spanned by necklaces, where the product퐴×퐵 is taken in the ∞-category Psh(횫)Δ0⊔Δ0/ of bipointed simplicial spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' By Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='19 the weak homotopy type of this category computes the mapping space |Nec/(퐴×퐵,(푎min,푏min),(푎max,푏max))| ≃ Map C(퐴×퐵) ((푎min,푏min), (푎max,푏max)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Since C(−) commutes with products by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='27, we may compute C(퐴 × 퐵) ≃ C(퐴) × C(퐵) = [푛] × [푚].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' In particular we see that the mapping space Map[푛]×[푚]((0, 0), (푛,푚)) is contractible, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' □ 2 Applications 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='1 Segalification and right fibrations Throughout this section we fix a presentable ∞-category V and a factorization system (V퐿, V푅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We say that 푋 : Δop → V is right-V푅-fibered if 푑0 : 푋푛 → 푋푛−1 is in V푅 for all 푛 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' A Segal object 푋 : 횫op → V is right-V푅-fibered if and only if 푑0 : 푋1 → 푋0 is in V푅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Indeed morphisms in V푅 are closed under pullbacks in the arrow category and when 푋 is Segal we can write 푑0 : 푋푛 → 푋푛−1 as a pullback in the arrow category of the following cospan 푋푛−1 푋0 푋1 푋푛−1 푋0 푋0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 푑1◦···◦푑푛 = = 푑0 = 푑1◦···◦푑푛 푑0 Under suitable assumptions the necklace formula can be used to show that Segalification preserves right-V푅-fibered objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 12 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Suppose V and (V퐿, V푅) are such that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' sifted colimits in V are stable under base change and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' V푅 /푋 ⊆ V/푋 is closed under sifted colimits for all 푋 ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Then Segalification Lseg : Fun(횫op, V) → Seg횫op (V) preserves right-V푅-fibered objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' By Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='2 it suffices to show that if 푋 : 횫op → V is right-V푅-fibered then 푑0 : L(푋)1 → L(푋)0 is in V푅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We claim that for every necklace 푁 = Δ푛1 ∨ · · · ∨ Δ푛푘 the map (푖∗푋)(푁) ≃ 푋푛1 ×푋0 · · · ×푋0 푋푛푘 → 푋0 induced by the inclusion of the terminal vertex Δ0 → 푁 is in V푅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Indeed, when 푁 is a simplex this holds by assumption and the general case follows by taking pullbacks since morphisms in V푅 are closed under pullbacks in the arrow category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Using the necklace formula (see Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='24), we can write 푑0 : L(푋)1 → L(푋)0 as a colimit L(푋)1 ≃ colim 푁 ∈Nec (푖∗푋)(푁) −→ 푋0 = L(푋)0 in V/푋0, of a diagram indexed by Necop, of morphisms (푖∗푋)(푁) → 푋0 that lie in V푅 /푋0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Since Necop is sifted (see Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='28) and V푅 /푋0 ⊆ V/푋0 is closed under sifted colimits, the colimit lies in V푅 /푋0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' □ Right fibrations of simplicial spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We shall now apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='3 to right fibrations of simplicial spaces in the sense of Rezk whose definition we briefly recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' A map of simplicial spaces 푝 : 푋• → 푌• is called a right fibration if the square 푋푛 푋푛−1 푌푛 푌푛−1 푑0 푝 푝 푑0 is cartesian for all 푛 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The Segalification formula implies the following: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' If 푝 : 푋 → 푌 is a right fibration, then so is the Segalification L(푝) : L(푋) → L(푌).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The target map 푡 : Ar(S) → S is a cartesian fibration and thus we have a factorization system on Ar(S) whose right part Ar(S)cart ⊆ Ar(S) consist of the cartesian edges, equivalently pullback squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' (The left part consists of morphisms which induce equivalence on the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=') Note that 푝 : 푋 → 푌 ∈ Ar(Psh(횫)) = Fun(횫op, Ar(S)) is right-Ar(S)cart-fibered if and only if it is a right fibration, hence it suffices to verify the conditions of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='3 for V = Ar(S) equipped with the aforementioned factorization system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The first condition holds since colimits in Ar(S) are universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The second condition holds since colimits in S are stable under base change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Examination of the proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='5 shows that the same result holds if S is replaced with any presentable ∞-category V in which sifted colimits are stable under base change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 13 Recall that the nerve N• : Cat∞ → Psh(횫) is fully faithful and its essential image is precisely the complete Segal spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We write C(−) : Psh(횫) → Cat∞ for the left adjoint of N•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We let LCSS: Psh(횫) → Psh(횫) denote the localization onto the complete Segal spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' With this notation we have for any 푋 ∈ Psh(횫) a canonical equivalence N• C(푋) ≃ LCSS푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' A model-categorical proof of the following proposition was given by Rasekh [Ras17, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='18 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The functor C: Psh(횫) → Cat∞ induces for any simplicial space 푋• an equivalence Psh(횫)r−fib /푋 C(−) −−−−→ ≃ Catr−fib ∞/C(푋) ≃ Psh( C(푋)) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Under the equivalence N• : Cat∞ ≃ CSeg횫op(S) the functor C(−) is identified with the localization LCSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='5 and [HK22, Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='21] the Segalification functor Lseg and the Rezk-completion functor L퐶 preserve right fibrations, so we have well-defined functors: LLCC : Psh(횫)r−fib /푋 Lseg −−−→ Psh(횫)r−fib /Lseg푋 L퐶 −−→ Psh(횫)r−fib /LCSS푋 The second functor is an equivalence by [HK22, Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='22] with inverse given by pullback along Lseg푋 → LCSS푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' It thus remains to show that the first functor is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Pulling back along 휑푋 : 푋 → Lseg푋 defines functor in the other direction: 휑∗ : Psh(횫)r−fib /푋 −→ Psh(횫)r−fib /Lseg푋 Note that a priori it is unclear whether Lseg and 휑∗ are adjoints of one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Nevertheless, we will prove they are inverse by showing that the composites in both directions are equivalent to the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' For the composite 휑∗ ◦ Lseg we start with a right fibration of simplicial spaces 푝 : 푋• → 푌• and we need to show that the comparison map 푋• → (푌• ×Lseg푌• LCSS푋•) is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Since this is a map of right fibrations over LCSS푌•, it suffices by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='8 that this is an equivalence on 0-simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' But on 0-simplices we have (Lseg푋)0 ≃ 푋0 and (Lseg푌)0 ≃ 푌0, so the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' For the composite Lseg ◦ 휑∗ we start with a right fibration 푝 : 퐸• → Lseg푌• over Lseg푌• and we need to show that the canonical dashed map in the diagram 휑∗ 푌퐸• Lseg(휑∗ 푌 퐸•) 퐸• 푌• Lseg푌• is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Again, this is a map of right fibrations over Lseg푌• and by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='8 it may be checked on 0-simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Using 푋0 ≃ (Lseg푋)0 twice we see that all horizontal maps in this diagram are equivalences, in particular the dashed one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Suppose 푋• → 푌• and 푋 ′ → 푌• are right fibrations and 푓 : 푋• → 푋 ′ is a map over 푌•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Then 푓 is an equivalence if and only if 푓0 : 푋0 → 푋 ′ 0 is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Since 푓 : 푋 ′ → 푋• is a map of right fibrations the map 푓푛 : 푋푛 → 푋 ′ 푛 can be recovered as the base change of 푓0 : 푋0 → 푋 ′ 0 along (푑0)푛 : 푋 ′ 푛 → 푋 ′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' □ 14 A formula for C(−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Let 푋 be a simplicial space and write 횫/푋 for its ∞-category of simplices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' the codomain of the associated right fibration 푝푋 : 횫/푋 → 횫.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='7 can be used to give a formula for C(푋) as a certain localization of 횫/푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' To do so we will need the “last vertex map” N•(횫/푋 ) → 푋 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' [HK22, §4]) and the “last vertex functor” 푒 : 횫/푋 → C(푋) obtained by applying C(−) to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Let us write 횫max ⊆ 횫 for the wide subcategory spanned by morphisms which preserve the maximal element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The "last vertex functor" 푒 : 횫/푋 → C(푋) induces an equivalence of ∞-categories 횫/푋 [푊 −1 푋 ] ≃ C(푋) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' where 푊푋 ≔ 푝−1 푋 (횫max) ⊆ 횫/푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The functor 푒 factors through 푒 : 횫/푋 [푊 −1 푋 ] → C(푋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Pulling back along 푒 induces functors Psh( C(푋)) −→ Psh(횫/푋 [푊 −1 푋 ]) ↩→ Psh(횫/푋 ) ≃ Psh(횫)/푋 and by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='7 the composite is fully faithful with image the right fibrations over 푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' By [BS22, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='8] the composite functor and the second functor have the same essential image and since the latter is fully faithful the first functor is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' By naturality of the Yoneda embedding it follows that 푒 is fully faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' It remains to verify that 푒 : 횫/푋 → C(푋) is essentially surjective and indeed the composite 푋0 → (횫/푋)≃ → C(푋)≃ ≃ (LCSS푋)0 is surjective on components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='2 Segalification for (∞,푛)-categories By iterating the Segalification formula one can also obtain formulas for the Segalification for (∞,푛)-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We recall the definition of 푛-fold Segal spaces due to Barwick [Bar].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' See [CS19, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='4] and [Hau18] for a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' An 푛-fold simplicial space 푋 : 횫op,푛 → S is reduced if for every 푘 ≥ 0 and푚1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' ,푚푘 ∈ N the (푛−푘 −1)-fold simplicial space 푋푚1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',푚푘,0,•,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',• is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We denote by Psh푟 (횫×푛) the full subcategory spanned by reduced objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' an 푛-uple Segal space if it is Segal in each coordinate, that is, if for every 푘 ≥ 0 and 푚1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' ,푚푘−1,푚푘+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' ,푚푛 ∈ N the simplicial space 푋푚1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',푚푘−1,•,푚푘+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',푚푛 is a Segal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' an 푛-fold Segal space if it is an 푛-tuple Segal space and reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We denote by Segn−fold 횫op ⊂ Psh푟 (횫×푛) the full subcategory spanned by the 푛-fold Segal spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' As we briefly explained in the introduction, complete 푛-fold Segal spaces model (∞,푛)-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We will not discuss the issue of completeness in this paper, but rather our goal is it give a formula for the Segalification of a reduced 푛-fold simplicial spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Given 1 ≤ 푗 ≤ 푛 we denote by L푗 : Psh(횫×푛) → Psh(횫×푛) the Segalification functor in the 푗-th coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 15 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Suppose 퐹 : 퐾 → Psh(횫) is a diagram of simplicial spaces such that 퐾 is sifted, each 퐹 (푘) is a Segal space, and the diagram 퐹 (−)0 : 퐾 → S is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Then the colimit colim 푘 ∈퐾 퐹 (푘) is a Segal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' A simplicial space 푋• is Segal if and only if the canonical map 푋푛 → 푋0 ×(푋0×푋0) (푋푛−1 × 푋1) is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' In the case of 푋• = colim 푘 ∈퐾 퐹 (푘)• we therefore want show that the outside rectangle in the following diagram is cartesian: colim 푘 ∈퐾 퐹 (푘)푛 colim 푘 ∈퐾 퐹 (푘)푛−1 × 퐹 (푘)1 colim 푘 ∈퐾 퐹 (푘)푛−1 × colim 푘 ∈퐾 퐹 (푘)1 colim 푘 ∈퐾 퐹 (푘)0 colim 푘 ∈퐾 퐹 (푘)0 × 퐹 (푘)0 colim 푘 ∈퐾 퐹 (푘)0 × colim 푘 ∈퐾 퐹 (푘)0 ≃ Δ ≃ The right horizontal maps are equivalences because 퐾 is sifted and hence it suffices to consider the left square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' This square is a colimit of the cartesian squares that we have because 퐹 (푘)• is Segal for all 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Moreover, the bottom row of the squares is a constant functor in 푘 by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' So it follows that the colimit of the square is still cartesian because colimits in S are universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' As observed above this implies that colim 푘 ∈퐾 퐹 (푘)• is Segal as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Let 푋•,···• be a reduced 푛-fold simplicial space, then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' (L푗푋)•,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',• is reduced for all 푗, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' if 푋•,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',• satisfies the Segal condition in the first (푗 − 1)-coordinates, then (L푗푋)•,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',• satisfies the Segal condition in the first 푗 coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Claim (1): We need to check that (L푗푋)푚1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',푚푘,0,•,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',• is a constant simplicial space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' For푘 < 푗 −1 this is true because constant simplicial spaces are Segal and hence Segalifying in the 푗th coordinate does not change 푋푚1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',푚푘,0,•,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' For 푘 = 푗 − 1 this is true because Segalifying never changes the 0-simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' For 푘 ≥ 푗 consider the simplicial object 푌 : 횫op → Psh(횫×{푘+2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',푛}) defined by sending 푙 to 푋푚1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',푚푗−1,푙,푚푗+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',푚푘,0,•,···•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' By assumption 푌푙 is a constant (푛 − 푘 − 1)-fold simplicial space for all 푙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Since the full subcategory of those (푛 − 푘 − 1)-fold simplicial spaces that are constant is closed under all colimits, it follows that (Lseg푌)푙 is still constant for all 푙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Claim (2): To simplify notation, we will assume that 푛 = 2 = 푗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' the general case is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Suppose that 푋•,• satisfies the Segal condition in the first coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' It suffices to show that L2푋•,• still satisfies the Segal condition in the first coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' In other words, we need to show that (L2푋)•,푙 is a Segal space for all 푙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' By the Segal condition in the second coordinate it suffices to do so for 푙 = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' For 푙 = 0 there is nothing to show since Segalification does not change the 0-simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' For 푙 = 1 we have the necklace formula (L2푋)•,1 ≃ colim Δ푛1∨···∨Δ푛푘 ∈Necop 푋•,푛1 ×푋•,0 · · · ×푋•,0 푋•,푛푘 where the pullbacks and colimit are computed in simplicial spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' To complete the proof it suffices to show that the diagram on the right hand side whose colimit we are taking satisfies the hypotheses of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The indexing category Necop is sifted by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='28 and each of the terms in the diagram is a Segal space since Segal spaces are closed under pullbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' It remains to 16 observe that the diagram of 0-simplices is constant since its value on any necklace 푁 = Δ푛1 ∨· · ·∨Δ푛푘 is 푋0,푛1 ×푋0,0 · · · ×푋0,0 푋0,푛푘 ≃ 푋0,0 ×푋0,0 · · · ×푋0,0 푋0,0 ≃ 푋0,0 by the reduced assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The left adjoint to the full inclusion of 푛-fold Segal space into reduced 푛-fold simplicial spaces may be computed as L = L푛 ◦ · · · ◦ L1 : Seg푛−fold 횫op ⇄ Psh푟 (횫×푛) :inc Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' For any 푛-fold simplicial space the map 푌•,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',• → (L푗푌)•,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',• is local with respect to those 푛-fold simplicial spaces that are Segal in the 푗th coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Therefore, for any 푛-fold simplicial space 푋•,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',• all of the maps 푋•,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',• → (L1푋)•,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',• → (L2 ◦ L1)(푋)•,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',• → · · · → (L푛 ◦ · · · ◦ L1)(푋)•,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',• are local with respect to the full subcategory of 푛-tuple Segal spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' If we assume that 푋•,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',• is also reduced, then it follows by inductively applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='12 that (L푗 ◦ · · · ◦ L1)(푋)•,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',• is reduced and satisfies the Segal condition in the first 푗 coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We have therefore shown that the map 푋•,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',• → (L푛 ◦ · · · ◦ L1)(푋)•,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=',• is local with respect to 푛-fold Segal spaces and that its target is an 푛-fold Segal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Consequently it exhibits the target as the localization onto the full subcategory Segn−fold 횫op .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' □ Warning 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' In the context of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='13 the order in which the Segalification functors are applied is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' It is important to Segalify the 1-morphisms first, then the 2-morphisms, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' If we were to apply L2 first and then L1 it would no longer clear that the result is L2-local as L1 can break the Segal condition in the second simplicial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='3 The Boardman-Vogt tensor product In this section we show how to use the Segalification formula to give a new construction of the Boardman-Vogt tensor product of ∞-operads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We begin with a brief recollection on the tensor product of commutative monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Recollection on tensor product of commutative monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' For an ∞-category with products C we let CMon(C) denote the ∞-category of commutative monoids in C, see [GGN16, §1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' In the case C = S we simply write CMon := CMon(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We let Cat⊗ ∞ := CMon(Cat∞) denote the ∞-category of symmetric monoidal ∞-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' By applying CMon(−) to the adjunction C(−) ⊣ N• and using that CMon(Psh(횫)) ≃ Fun(횫op, CMon) we obtain an adjunction: C(−) : Fun(횫op, CMon) ⇄ Cat⊗ ∞ :N•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The right adjoint here is the symmetric monoidal nerve, which in the 푛th level is given by N푛(D) = Fun([푛], D)≃ with the pointwise symmetric monoidal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 17 Let C be a cartesian closed presentable ∞-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Gepner, Groth and Nikolaus [GGN16] show that CMon(C) admits a canonical symmetric monoidal structure ⊗ such that the free functor F: C → CMon(C) is symmetric monoidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Moreover, if C and D are presentable cartesian closed and 퐿: C → D is a symmetric monoidal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' finite product preserving) left adjoint they show the induced functor 퐿: CMon(C) → CMon(D) is canonically symmetric monoidal [GGN16, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='(ii)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Segalification, completion, and C(−) are examples of such 퐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We record this for future use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' All of the functors in the following commutative diagram are canonically symmetric monoidal for the respective tensor product: Fun(횫op, CMon) Seg횫op(CMon) CSeg횫op(CMon) Cat⊗ ∞ L퐶 Lseg C(−) ≃ N• (−) Some equifibered theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' A morphism of commutative monoids 푓 : 푀 → 푁 is said to be equifibered if the canonical square 푀 × 푀 푀 푁 × 푁 푁 푓 ×푓 + + 푓 is cartesian [BS22, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' This notion was introduced in op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' for the purpose of developing the theory of ∞-properads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' A quintessential feature of equifibered maps is that a morphism of free monoids 푓 : F(푋) → F(푌) is equifibered if and only if it is free, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 푓 ≃ F(푔) for some map of spaces푔: 푋 → 푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Equifibered maps can be thought of as a well-behaved generalization of free maps, for example they form the right part of a factorization system on CMon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Further details can be found in [BS22, §2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Equifibered maps between free monoids are closed under tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Indeed, the free functor F: S → CMon is symmetric monoidal and by [BS22, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='7] induces an equivalence onto the subcategory CMonfree,eqf ⊆ CMon of free monoids and equifibered maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' A simplicial commutative monoid 푀 : 횫op → CMon is called ⊗-disjunctive if it is right-CMoneqf-fibered where CMoneqf ⊆ CMon denotes the subcategory of equifibered mor- phisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' By [BS22, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='15] the nerve N•C of a symmetric monoidal ∞-category C ∈ Cat⊗ ∞ is ⊗-disjunctive, if and only if C is ⊗-disjunctive in the sense of [BS22, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' That is, if and only if for all 푥,푦 ∈ C the functor ⊗: C/푥 × C/푦 −→ C/푥 ⊗푦 is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Let 푀 : 횫op → CMon be ⊗-disjunctive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Then 푀 is level-wise free if and only if 푀0 is free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Indeed, evaluation at the last vertex 푑0 ◦ · · · ◦ 푑0 : 푀푛 → 푀0 is equifibered so if 푀0 is free the same holds for 푀푛 [BS22, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' As a consequence of the necklace formula we have the following: 18 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Let 푀 ∈ Fun(횫op, CMon) be ⊗-disjunctive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Then Lseg(푀) is ⊗-disjunctive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' It suffices to check the conditions of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The first condition was verified in Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The second condition follows from [BS22, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' □ We now give a description of ∞-operads using equifibered maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' A pre-operad is a Segal commutative monoid 푀 ∈ Seg횫op (CMon) which is ⊗- disjunctive and level-wise free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' A pre-operad is called complete if its underlying Segal space is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Warning 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Pre-operads in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='21 should not be confused with ∞- preoperads in the sense of Lurie [Lur, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Instead, in the language of [BS22], a pre-operad is precisely a monic pre-properad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Pre-operads are to ∞-operads what Segal spaces are to ∞-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Indeed the envelope functor induces an equivalence between Lurie’s ∞-operads and complete pre-operads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Lurie’s monoidal envelope functor Env(−) : Op∞ → Cat⊗ ∞ is faithful (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' it induces a monomorphism on mapping spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Moreover, the composite Op∞ Env(−) −−−−−→ Cat⊗ ∞ N•≃ CSeg(CMon) identifies Op∞ with the (non-full) subcategory of CSeg(CMon) whose objects are complete pre-operads and whose morphisms are equifibered natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The first instance of this theorem can be found in the work of Haugseng–Kock [HK21], who showed that the sliced functor Env: Op∞ → Cat⊗ ∞/Fin is fully faithful and characterised its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Barkan– Haugseng–Steinebrunner [BHS22] then gave an alternative characterization of the image, closely related to pre-operads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The above formulation was given in [BS22, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Tensor product of ∞-operads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' We can now apply the Necklace formula for Segalification to show that pre-operads are closed under tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The replete subcategory pOp∞ ⊆ Seg횫op (CMon) is closed under tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' First we claim that if 푀 : 횫op → CMon is level-wise free and ⊗-disjunctive then the same holds for Lseg푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Indeed Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='20 shows that Lseg(푀) is ⊗-disjunctive and since Lseg(푀)0 ≃ 푀0 is free the claim follows from Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' To complete the proof it suffices to show that if 푀, 푁 ∈ Fun(횫op, CMon) are ⊗-disjunctive and level-wise free, then the same holds for their tensor product 푀 ⊗ 푁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' This follows from the fact that equifibered maps between free monoids are closed under tensor product (see Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' □ We are now in a position to prove Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 19 Proof of Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' For the first part it suffices by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='23 to show that complete pre-operads are closed under the tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='15 the equivalence Cat⊗ ∞ ≃ CSeg횫op (CMon) identifies the tensor product of symmetric monoidal ∞-categories with the following bi-functor on complete Segal monoids (푀•, 푁•) ↦−→ LCSS(푀• ⊗ 푁•) ≃ L퐶Lseg(푀• ⊗ 푁•).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Suppose now that 푀• and 푁• are pre-operads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='24, Lseg(푀• ⊗ 푁•) is a pre-operad and hence [BS22, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='7] so is the completion L퐶Lseg(푀• ⊗ 푁•).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' For the second part we must compare ⊗BV to Lurie’s tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' It follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='25 that there is an equivalence Env(O ⊗Lurie P) ≃ Env(O ⊗BV P) for all ∞-operads O and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' And since Env is an equivalence onto a replete subcategory by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='23 it follows that we already have such an equivalence before applying Env.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Lurie’s tensor product satisfies that for any two ∞-operads O and P there is a canonical equivalence: Env(O ⊗Lurie P) ≃ Env(O) ⊗ Env(P) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The tensor product in Cat⊗ ∞ is adjoint to the internal hom: FunCat⊗ ∞ (C ⊗ D, E ) ≃ FunCat⊗ ∞(C, FunCat⊗ ∞ (D, E )), and monoidal envelope Env: Op∞ → Cat⊗ ∞ is left adjoint to the forgetful functor Cat⊗ ∞ → Op∞ and by [Lur, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='] we have an equivalence: FunCat⊗ ∞ (Env(O), E ) ≃ AlgO(E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Combining these facts with the key property5 of Lurie’s tensor product that AlgO⊗LurieP (E ) ≃ AlgO(AlgP (E )) we obtain a sequence of equivalences: FunCat⊗ ∞(Env(O ⊗Lurie P), E ) ≃ AlgO⊗LurieP (E ) ≃ AlgO(AlgP (E )) ≃ FunCat⊗ ∞ (Env(O), FunCat⊗ ∞(Env(P), E )) ≃ FunCat⊗ ∞ (Env(O) ⊗ Env(P), E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' As this is natural in E ∈ Cat⊗ ∞ the claim follows from the Yoneda lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' □ References [Bar] Clark Barwick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' (∞,푛)-Cat as a closed model category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' PhD thesis, unpublished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' [BHS22] Shaul Barkan, Rune Haugseng, and Jan Steinebrunner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Envelopes for Algebraic Patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Available at arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='07183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' [BR20] Julia E Bergner and Charles Rezk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' “Comparison of models for (∞,푛)-categories, II”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' In: Journal of Topology 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='4 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 1554–1581.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 5This follows from the definition of the symmetric monoidal structure on AlgP (E ) [Lur, Construction 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='1] and the definition of Lurie’s BV tensor product [Lur, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 20 [BS21] Clark Barwick and Christopher Schommer-Pries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' “On the unicity of the homotopy theory of higher categories”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' In: Journal of the American Mathematical Society 34 (2021), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 1011–1058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' [BS22] Shaul Barkan and Jan Steinebrunner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' The equifibered approach to ∞-properads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Available at arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='02576.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' [BV73] John Michael Boardman and Rainer M Vogt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Homotopy invariant algebraic structures on topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 347.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Springer, 1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' [CS19] Damien Calaque and Claudia Scheimbauer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' “A note on the (∞,푛)-category of cobor- disms”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' In: Algebraic and Geometric Topology 19 (2019), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 533–655.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' [DS11] Daniel Dugger and David I Spivak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' “Rigidification of quasi-categories”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' In: Algebraic and Geometric Topology 11 (2011), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 225–261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' [GGN16] David Gepner, Moritz Groth, and Thomas Nikolaus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' “Universality of multiplicative infinite loop space machines”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' In: Algebraic & Geometric Topology 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='6 (2016), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 3107– 3153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' [Hau18] Rune Haugseng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' “On the equivalence between Θ푛-spaces and iterated Segal spaces”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' In: Proceedings of the American Mathematical Society 146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='4 (2018), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 1401–1415.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' [Hau21] RuneHaugseng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' “Onlaxtransformations,adjunctions,andmonadsin (∞, 2)-categories”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' In: Higher Structures 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='1 (2021), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 244–281.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' [Hau22] RuneHaugseng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' “∞-Operadsvia symmetricsequences”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' In:Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='301 (2022),pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='115– 171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' [HK21] Rune Haugseng and Joachim Kock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' ∞-operads as symmetric monoidal ∞-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' To appear in Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Available at arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='12975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' [HK22] Philip Hackney and Joachim Kock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Culf maps and edgewise subdivision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Available at arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='11191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' [Joy07] AndréJoyal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Notesonquasicategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' URL:http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='utoronto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='ca/av/slides/06-07/crs-quasibasic/joyal/download.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' [JT06] André Joyal and Myles Tierney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' “Quasi-categories vs Segal spaces”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' In: Categories in Algebra, Geometry and Mathematical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 2006, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 277–326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' [Lur] Jacob Lurie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' Higher Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' url: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='ias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='edu/~lurie/papers/HA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' [Lur09] Jacob Lurie.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' In: Transactions of the American Mathematical Society 353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='3 (2001), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 973–1007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' [Rez10] Charles Rezk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' “A Cartesian presentation of weak n–categories”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' In: Geometry & Topology 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content='1 (2010), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 521–571.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' [Ste17] Danny Stevenson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' “Covariant model structures and simplicial localization”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' In: North- Western European Journal of Mathematics 3 (2017), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFAT4oBgHgl3EQfrB0d/content/2301.08650v1.pdf'} +page_content=' 141–202.' metadata={'source': 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Science and Technology, Nanjing +210094, China +2College of Electronic and Optical Engineering & College of Flexible Electronics (Future Technology), +Nanjing University of Posts and Telecommunications, Nanjing 210023, China +3Collaborative Innovation Center of Advanced Microstructures and School of Physics, Nanjing +University, Nanjing 210093, China +4jpding@nju.edu.cn +*zhangbf@njust.edu.cn +Abstract: The Pancharatnam–Berry (PB) phase in metasurfaces obeys the symmetry restriction, +according to which the PB phases of two orthogonal circularly polarized waves are the same +but with opposite signs. Here, we reveal a general mechanism to break the axisymmetry of +meta-atoms in order to break the PB-phase symmetry restriction. As a proof of concept, we +designed a novel meta-atom with a QR-code structure and successfully demonstrated circular- +polarization multiplexing metasurface holography. This study provides a fundamentally new +understanding of the PB phase and opens a path for arbitrary wavefront engineering using +asymmetric electromagnetic structures. + +1. Introduction +Metasurfaces, as ultrathin electromagnetic (EM) functional layers, have emerged as a +comprehensive and compact platform for wavefront engineering in the last decade [1,2]. The +overall wavefront engineering by a metasurface results from the linear superposition of EM +waves modulated by each meta-atom, which acts as the unit cell of the metasurface. The +conventional phase-modulation mechanisms of a meta-atom include the dynamic phase and +Pancharatnam–Berry (PB) phase (or geometric phase). The dynamic phase originates from the +accumulated phases of an EM wave propagating in a meta-atom; the phase accumulation +depends on the refractive index, geometry, and other meta-atom parameters [3,4]. The PB phase +arises from the spin–orbit coupling of photons in a meta-atom, as a function of the rotation +angle of the anisotropic EM structures (including elliptical structures and rectangular structures) +[5–7]. For example, when a circularly polarized EM wave propagates through such structures +with a rotation angle θ, the transmitted cross-polarized component adopts the so-called PB +phase (Φ). Conventionally, Φ = ±2θ, where the sign ± is determined by the chirality of the EM +wave; for instance, the left- and right-circular polarizations (LCP and RCP, respectively). +Unlike the dynamic phase, the PB phase is symmetric for both the orthogonal circular +polarizations. In other words, the PB phases of the LCP and RCP have the same absolute value +but opposite signs. Recently, Xie et al. studied the rotational symmetry of meta-atoms and +demonstrated high-order PB phases, which are equivalent to several times the rotation angle +(rather than just two times) [8]. However, the PB phase of these meta-atoms remains symmetric. +Therefore, achieving polarization-decoupled EM wavefront engineering using only the +geometric phase is difficult. +In recent years, some researchers attempted to break the symmetry of the PB phase. For +example, by introducing nonlinear effects, the PB phase of transmitted harmonic waves can be +rewritten in the form of ±(n±1) θ, where n is the order of harmonic generation [9–11]. However, + +this method cannot be used to engineer the PB phases of orthogonal circular polarizations +arbitrarily. In the case of linear processes, Bai et al. proposed to combine the Aharonov– +Anandan and PB phases to break the symmetry limitation of the PB phase [12]. This proposed +method can be applied in EM wave engineering metasurfaces [13,14]. Chen et al. proposed a +type of planar chiral meta-atom to perform local phase manipulations in metasurfaces and +realized a controlled spin decoupling of the orthogonal circularly polarized waves. [15]. The +studies reported to date trace the origins of asymmetric PB phases from different physical +mechanisms and demonstrate polarization-decoupled PB phase modulation. However, the +identification of a general fundamental factor that can break the symmetry limitation of the PB +phase for orthogonal circularly polarized waves is crucial for understanding the relationship +between meta-atom structures and their corresponding EM wavefront engineering properties. +In this study, we analytically demonstrated that the symmetry restriction of the PB phase +originates from the axisymmetry of EM structures for the first time to the best of our knowledge. +In other words, from the geometrical and topological point of view, a non-axisymmetric meta- +atom topology naturally leads to symmetry breaking of the PB phases of two orthogonal +circularly polarized waves, although different geometric structures may have different +polarization-decoupling mechanisms. Further, we analytically established the relationship +between the Jones matrices based on the circular- and linear-polarization base vectors. +Moreover, we designed and investigated non-axisymmetric meta-atoms with QR-code +structures to verify the breaking of the symmetry restriction of the PB phase. As a proof of +concept, we designed a circularly polarized multiplexed hologram using a metasurface +consisting of QR-code meta-atoms and numerically demonstrated our proposed theory. The +results of this study provide a fundamentally new understanding of the PB phase and light– +matter interactions in nanophotonics and can thus promote more advanced metasurface- and +metamaterial-based applications. +2. Theory of PB phase for non-axisymmetric meta-atoms +First, we derive a general form of the PB phase for a meta-atom with an arbitrary topology, e.g., +a non-axisymmetric structure. Note that the structure of a three-dimensional (3D) meta-atom is +also called a “non-mirror symmetric” structure instead of a non-axisymmetric structure. +However, considering that the topologies of meta-atoms usually vary in the top-view plane and +remain unchanged in the height dimension, we perform the analysis using the “non- +axisymmetry” concept for simplicity but without loss of generality. For an EM wave incident +on a meta-atom, the relationship between the transmitted and incident waves can be established +using the Jones matrix. Usually, we use linearly polarized base vectors to describe the incident +and transmitted EM waves as follows: +𝐸𝑖𝑛 = 𝑎1𝑥̂ + 𝑎2𝑦̂ = (𝑎1 +𝑎2), (1) +𝐸𝑜𝑢𝑡 = 𝐽𝐸𝑖𝑛 = (𝐽11 +𝐽12 +𝐽21 +𝐽22) (𝑎1 +𝑎2), (2) +where Ein and Eout are the incident and transmitted EM waves, respectively; 𝑥̂ and 𝑦̂ are the +base vectors of the x and y linear polarizations, respectively; a1 and a2 are the corresponding +complex amplitudes with 𝑎1 +2 + 𝑎2 +2 = 1 as the normalization condition; and J is the Jones matrix +for linear-polarization base vectors. For an arbitrary meta-atom structure, the four elements of +J are all non-zero values. However, for an axisymmetric structure, J21 = J12 = 0 (details in the +Appendix), which suggests that a linear-polarization-decoupled metasurface can be designed +using axisymmetric meta-atoms as discussed in our previous work [16]. +Introducing circular-polarization base vectors to the above equations, we obtain + +𝐸𝑖𝑛 = 𝑏1𝐿̂ + 𝑏2𝑅̂ = (𝑏1 +𝑏2), (3) +𝐸𝑜𝑢𝑡 = 𝑆𝐸𝑖𝑛 = (𝑆11 +𝑆12 +𝑆21 +𝑆22) (𝑏1 +𝑏2), (4) +where 𝐿̂ and 𝑅̂ are the base vectors of LCP and RCP, respectively; b1 and b2 are the +corresponding complex amplitudes with 𝑏1 +2 + 𝑏2 +2 = 1 as the normalization condition; and S is +the Jones matrix for circular-polarization base vectors. If we further consider the relationship +between two sets of orthogonal base vectors as 𝐿̂ = √2 +2 (𝑥̂ − 𝑖𝑦̂) and 𝑅̂ = √2 +2 (𝑥̂ + 𝑖𝑦̂), and +conduct some analytical derivations (details in the Appendix), then we can rewrite the S matrix +in the form of J matrix as follows: +{ + + + + 𝑆11 = +1 +2 [(𝐽11 + 𝐽22) − 𝑖(𝐽12 − 𝐽21)] +𝑆21 = +1 +2 [(𝐽11 − 𝐽22) − 𝑖(𝐽12 + 𝐽21)] +𝑆12 = +1 +2 [(𝐽11 − 𝐽22) + 𝑖(𝐽12 + 𝐽21)] +𝑆22 = +1 +2 [(𝐽11 + 𝐽22) + 𝑖(𝐽12 − 𝐽21)] + . (5) +Further, if the meta-atom rotates by an angle θ, then the Jones matrix is expressed as +𝐽(𝜃) = 𝑅(−𝜃)𝐽𝑅(𝜃) = (𝑐𝑜𝑠𝜃 +−𝑠𝑖𝑛𝜃 +𝑠𝑖𝑛𝜃 +𝑐𝑜𝑠𝜃 ) (𝐽11 +𝐽12 +𝐽21 +𝐽22) ( 𝑐𝑜𝑠𝜃 +𝑠𝑖𝑛𝜃 +−𝑠𝑖𝑛𝜃 +𝑐𝑜𝑠𝜃). (6) +We substitute Eqs. (2) and (6) into Eq. (5), and finally obtain the most general form of the PB +phase for an arbitrary meta-atom using circular-polarization base vectors as follows: +{ + + + + 𝑆11(𝜃) = +1 +2 [(𝐽11 + 𝐽22) − 𝑖(𝐽12 − 𝐽21)] +𝑆21(𝜃) = +1 +2 [(𝐽11 − 𝐽22) − 𝑖(𝐽12 + 𝐽21)]𝑒−𝑖2𝜃 +𝑆12(𝜃) = +1 +2 [(𝐽11 − 𝐽22) + 𝑖(𝐽12 + 𝐽21)]𝑒𝑖2𝜃 +𝑆22(𝜃) = +1 +2 [(𝐽11 + 𝐽22) + 𝑖(𝐽12 − 𝐽21)] +. (7) +Compared to the linear Jones matrix J(θ), S(θ) intuitively exhibits numerous important +properties. First, the diagonal elements S11 and S22 remain unchanged even if the meta-atom is +rotated. Second, the anti-diagonal elements S21 and S12 for the transmitted cross-polarizations +are naturally symmetry-broken. In other words, the cross-polarization transmissions of the +incident circularly polarized EM waves are inherently asymmetric, which is contrary to the +traditional view. PB phases are subjected to symmetry restrictions in conventional meta-atoms +such as those with rectangular, elliptical, and other structures, which have been widely studied +in previous works, because for the aforementioned axisymmetric topologies, J12 = J21 = 0. Thus, +the asymmetric elements S21(θ) and S12(θ) degenerate into the most known forms with a +symmetric PB phase: +{ +𝑆21(𝜃) = +1 +2 (𝐽11 − 𝐽22)𝑒−𝑖2𝜃 +𝑆12(𝜃) = +1 +2 (𝐽11 − 𝐽22)𝑒𝑖2𝜃 . (8) +The conventional symmetry restriction of the PB phase reportedly originates from the +axisymmetry of meta-atoms, and thus breaking the axisymmetry of the meta-atoms intrinsically +leads to breaking of the PB-phase symmetry. Notably, instead of the structural symmetry, the +optical mode (electromagnetic field) inside the meta-atom structure should be adopted as a +more general physical quantity in this case; however, these two factors are usually consistent +with each other. Thus, for simplicity, we perform our analysis from the structural point of view. + +Next, we rewrite S21(θ) and S12(θ) in Eq. (7) in a more intuitive form. If S21 and S12 simplify +to 𝐴21𝑒𝑖𝜑21 and 𝐴12𝑒𝑖𝜑12, respectively, then S21(θ) and S12(θ) can be expressed as: +{𝑆21(𝜃) = (𝐴21𝑒𝑖𝜑12)𝑒−𝑖(2𝜃−∆𝜑) +𝑆12(𝜃) = (𝐴12𝑒𝑖𝜑12)𝑒𝑖2𝜃 +, (9) +where A12 and A21 are the amplitudes of S21 and S12, respectively; 𝜑21 and 𝜑12 are the argument +angles of S21 and S12, respectively; and ∆𝜑 = 𝜑21 − 𝜑12. The symmetry of PB phase is broken +by the key phase ∆𝜑. +3. Schematic of the non-axisymmetric meta-atom with QR-code structure +In order to break the symmetry restriction of the PB phase, the meta-atom should break the +axisymmetry in the topology. Here, we design a novel dielectric meta-atom with a QR-code +structure to eliminate any geometric symmetry as shown in Fig. 1. This type of QR-code +topology, which was investigated in our previous study on perfect metamaterial absorbers, +possesses a large number of degrees of freedom and can thus provide rich properties for +wavefront engineering [17]. In this study, the meta-atom, with a period P of 800 nm, consists +of a silica substrate and arrays of square silicon pillars that appear as a QR code. Each QR code +consists of 5 × 5 pixels, and each pixel is randomly set as either a square silicon pillar (length: +100 nm; height: 1400 nm) or air. Because the generated QR-code topology is completely +random, it easily and thoroughly breaks the axisymmetry. + +Fig. 1 Schematic of the non-axisymmetric meta-atom with QR-code structure: (a) 3D view and (b) top view. The +parameters are set as P = 800 nm, L = W = 100 nm, and H = 1400 nm. +We conducted full-wave 3D finite-difference time-domain (FDTD) simulations to design +and investigate the QR-code meta-atom structures. In the simulations, we set the refractive +index of the silica substrate as 1.45, and derived the dielectric constant values of silicon by +fitting Palik’s experiment data to the Drude–Lorentzian model [18,19]. For the meta-atom +simulation, we used the period boundaries in the horizontal direction and perfectly matched +layers (PMLs) in the longitudinal direction. By randomly setting each pixel of the QR-code +structure, we obtained a variety of non-axisymmetric meta-atoms and numerically calculated +their parameters. All the simulated meta-atoms show asymmetric PB phases, which are +discussed in detail in the next section. +4. Breaking the symmetry restriction of PB phase using non-axisymmetric +meta-atoms +We designed and numerically evaluated 13,000 meta-atom structures with randomly arrayed +5×5 QR-code nanopillars. All the QR-code meta-atoms show decoupled PB phases for LCP +and RCP. We selected structures with transmission coefficients higher than 0.7 to ensure +efficient polarization-multiplexing applications, and the PB phases of these selected structures +are plotted in Fig. 2(a). Evidently, the PB phases of the LCP and RCP cover the 0–2π range and + +(a) +(b) +Si +SiOz +Pbreak the symmetry restriction, consistent with the predictions of Eqs. (7) and (9). For +simplicity but without loss of generality, we do not rotate the meta-atoms in Fig. 2, indicating +that the rotation angle θ is zero. The corresponding results suggest that the polarization- +decoupled PB phases, which originate from non-axisymmetric structures, are new degrees of +freedom for engineering EM wavefronts. In addition, we can independently engineer the PB +phases for orthogonal circular polarizations without rotating the meta-atoms in the full range +from 0 to 2π. + +Fig. 2 PB phases of different meta-atom structures. (a) PB phases of the QR-code structures shown in Fig. 1. (b) PB +phases of different axisymmetric structures. +We investigated a variety of axisymmetric meta-atom structures as a control group, and +their PB phases are plotted in Fig. 2(b). All axisymmetric meta-atoms have the same period +(800 nm) and height (1400 nm) as the non-axisymmetric ones, but different topologies in the +top view. All PB phases are located on the y = x line, irrespective of the meta-atom topology +(such as rectangle, ellipse, trapezoid, cross, T-shape, H-shape, or hollow rectangle), implying +that all PB phases of the two orthogonal circular polarizations have the same value. This is the +symmetry restriction of conventional PB phases, which is predicted in Eq. (8), and originates +from axisymmetric structures. +From Fig. 2, we can verify our analytical predictions of the PB phases; that is, decoupling +of the PB phases can be achieved under a broken axial symmetry. Furthermore, the decoupled +PB phase, independent of the rotation angle of the meta-atom, provides a new degree of +freedom in wavefront engineering. As a proof of concept, we designed a circular-polarization +multiplexing metasurface hologram based on our proposed QR-code meta-atoms. +The design of the circular-polarization-decoupled metasurface hologram is illustrated in a +flowchart in Fig. 3. First, we calculated the phase-only computer-generated holograms (CGHs) +for left- and right-circular polarized incident light beams using the Gerchberg–Saxton (GS) +algorithm [20]. We coded two different images for the two orthogonal polarizations and +obtained two output holographic phase diagrams after the GS iterations. Second, we performed +an eight-level phase quantization to rewrite the two phase diagrams into two phase matrices. In +this process, we selected 8 × 8 meta-atom structures from Fig. 2(a) to establish the eight-level +phase library, which was then searched to identify the meta-atom structures that fulfilled both +the phase requirements for the LCP and RCP holograms; to identify these structures, a one-by- +one element search of the phase matrices was performed. Finally, we simulated the desired +metasurface hologram with 64 types of QR-code structures. Due to the limited computing +resources, we could simulate a metasurface with 101 × 101 periods as a proof of concept, which +successfully demonstrated our theory. As the period was set to 800 nm, the metasurface +hologram could cover an area of 80.8 × 80.8 μm2. +Phase of S21 (unit: π) +Phase of S12 (unit: π) +(a) +Phase of S21 (unit: π) +Phase of S12 (unit: π) +(b) + + +Fig. 3 Design flowchart of the circular-polarization-decoupled metasurface hologram. + + +Fig. 4. Circular-polarization-decoupled metasurface holography. (a) and (e) are the original images for LCP and RCP +illuminations, respectively. (b) and (f) correspond to the phase distributions of the calculated holograms. (c) and (g) +are the images of the theoretical reconstructions. (d) and (h) correspond to the reconstructed images of the simulated +metasurface hologram. +The designed metasurface hologram was simulated using 3D FDTD and is shown in Fig. 3, +and the reconstructed images for LCP and RCP are presented in Fig. 4. Figures 4 (a) and (e) +show the 101 × 101-pixel dog and cat images as the original images, which are illuminated by +left- and right-circularly polarized light, respectively. Thus, the images reconstruct using the +PB phases incorporate the cross-polarization components RCP and LCP. The phase-only CGHs +of the two orthogonal polarizations, simulated using the GS algorithm, are show in Figs. 4(b) +and (f). As the control group, the images of the theoretical reconstructions are shown in Figs. +4(c) and (g). The reconstructed images of the simulated metasurface hologram, which was +illuminated by different circularly polarized EM waves, are plotted in Figs. 4(d) and (h). We +can see that the reconstructed images of the metasurface hologram show good agreement with +the original and theoretically reconstructed images, thereby confirming the feasibility of +decoupling circularly polarized light via wavefront engineering. This result demonstrates that +non-symmetric meta-atom structures can indeed break the symmetry restriction of PB phases +and can provide new insights into metasurfaces and metamaterials to promote their application +in various fields. +5. Conclusion +In conclusion, we theoretically established a general formula to describe the wavefront +engineering property of a meta-atom using the Jones matrix and circular-polarization base +vectors. The analytical result indicates that the non-axisymmetry of meta-atom leads to the +breaking of the symmetry restriction of the PB phase. As an illustrative example, we designed +π +0 +-π +1 +0.5 +0 +π +0 +-π +1 +0.5 +0 +1 +0.5 +0 +1 +0.5 +0 +(a) +(e) +(b) +(f) +(c) +(g) +(d) +(h) + +Design of CGH +8-level-phase library +Label +Plharse +Designed metasurface +心 +LCP +1/8-2x +GS I +Image +-8Z +Iteration +3/8 -2x +E +4/8 -2元 +RCP +5/8 -2元 +Image +.5 +6/8-2# +7/8-2元a novel QR-code meta-atom to break the topological axisymmetry and realize polarization- +decoupled PB-phase engineering. Further, we designed a circularly polarized multiplexed +metasurface hologram using our proposed QR-code meta-atoms to reveal the potential of +symmetry-broken PB phases in various applications in metasurfaces and metamaterials. We +believe that this study will expand the understanding of the PB phase in a fundamental way and +will facilitate the development of design methodologies for EM structures, which can be used +for arbitrary wavefront engineering. + +Appendix A: Derivation of Jones matrix S +When a left-circularly polarized light beam is incident on an arbitrary meta-atom, the +transmitted (or reflected) light is +𝐸𝑜𝑢𝑡 = 𝑆𝐸𝑖𝑛 = (𝑆11 +𝑆12 +𝑆21 +𝑆22) (1 +0) = 𝑆11𝐿̂ + 𝑆21𝑅̂ = 𝑆11 +√2 +2 (𝑥̂ − 𝑖𝑦̂) + 𝑆21 +√2 +2 (𝑥̂ + 𝑖𝑦̂) (A1) +Similarly, when a right-circularly polarized light beam incidents, the output light field is +expressed as: +𝐸𝑜𝑢𝑡 = 𝑆𝐸𝑖𝑛 = (𝑆11 +𝑆12 +𝑆21 +𝑆22) (0 +1) = 𝑆12𝐿̂ + 𝑆22𝑅̂ = 𝑆12 +√2 +2 (𝑥̂ − 𝑖𝑦̂) + 𝑆22 +√2 +2 (𝑥̂ + 𝑖𝑦̂) (A2) +We can rewrite the above equations using linear-polarization base vectors. For a left-circularly +polarized incident light beam, the transmitted electric field is given by +𝐸𝑜𝑢𝑡 = 𝐽𝐸𝑖𝑛 = (𝐽11 +𝐽12 +𝐽21 +𝐽22) ( +√2 +2 +− √2 +2 𝑖 +) = √2 +2 (𝐽11 − 𝑖𝐽12)𝑥̂ + √2 +2 (𝐽21 − 𝑖𝐽22)𝑦̂ (A3) +Similarly, for a right-circularly polarized incident light beam, we obtain +𝐸𝑜𝑢𝑡 = 𝐽𝐸𝑖𝑛 = (𝐽11 +𝐽12 +𝐽21 +𝐽22) ( +√2 +2 +√2 +2 𝑖 +) = √2 +2 (𝐽11 + 𝑖𝐽12)𝑥̂ + √2 +2 (𝐽21 + 𝑖𝐽22)𝑦̂ (A4) +The physical processes represented by the aforementioned equations remain identical +irrespective of using a circular- or linear-polarization base vector. Thus, we solve Eqs. (A1)– +A(4) simultaneously and establish the analytical relationship between J and S as follows: +{ + + + + 𝑆11 = +1 +2 [(𝐽11 + 𝐽22) − 𝑖(𝐽12 − 𝐽21)] +𝑆21 = +1 +2 [(𝐽11 − 𝐽22) − 𝑖(𝐽12 + 𝐽21)] +𝑆12 = +1 +2 [(𝐽11 − 𝐽22) + 𝑖(𝐽12 + 𝐽21)] +𝑆22 = +1 +2 [(𝐽11 + 𝐽22) + 𝑖(𝐽12 − 𝐽21)] + (A5) +Appendix B: Axisymmetry of meta-atoms +For an x-polarized incident light beam, the output light wave is given by +(𝐽11 +𝐽12 +𝐽21 +𝐽22) (1 +0) = 𝐽11𝑥̂ + 𝐽21𝑦̂ (B1) +If a mirror operation is performed on the meta-atom along the symmetric axis (for example, +the x-axis), then the Jones matrix 𝐽′ becomes (𝐽11 +′ +𝐽12 +′ +𝐽21 +′ +𝐽22 +′), and the output becomes + +(𝐽11 +′ +𝐽12 +′ +𝐽21 +′ +𝐽22 +′ ) (1 +0) = 𝐽11 +′ 𝑥̂ + 𝐽21 +′ 𝑦̂ (B2) +According to the property of mirror operation, the y-polarization component effectively +undergoes a phase retardation of e𝑖𝜋, and thus, we obtain 𝐽11𝑥̂ = 𝐽11 +′ 𝑥̂, 𝐽21𝑦̂ = 𝐽21 +′ 𝑦̂ ∙ 𝑒𝑖𝜋, +which implies 𝐽11 +′ = 𝐽11，𝐽21 +′ = −𝐽21. +Similarly, in the case of the y-polarization incident light beam, we can obtain 𝐽12 +′ = −𝐽12， +𝐽22 +′ = 𝐽22. Thus, the relationship between J and 𝐽′ can be expressed as +𝐽′ = (𝐽11 +′ +𝐽12 +′ +𝐽21 +′ +𝐽22 +′ ) = ( 𝐽11 +−𝐽12 +−𝐽21 +𝐽22 ) (B3) +For an axisymmetric meta-atom, mirror operation on its symmetric axis does not change its +geometry. Thus, we obtain following equation: +𝐽′ = (𝐽11 +′ +𝐽12 +′ +𝐽21 +′ +𝐽22 +′ ) = ( 𝐽11 +−𝐽12 +−𝐽21 +𝐽22 ) = (𝐽11 +𝐽12 +𝐽21 +𝐽22) = 𝐽 (B4) +which indicates 𝐽12 = −𝐽12 = 0，𝐽21 = −𝐽21 = 0. Thus, the Jones matrix J of the +axisymmetric meta-atoms is simplified to +𝐽𝑎𝑥𝑖𝑠−𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑦 = (𝐽11 +0 +0 +𝐽22) (B5) +Further, if the axisymmetric meta-atom rotates by an angle θ, then the corresponding Jones +matrix S(θ) in Eq. (7) is simplified to +𝑆𝑎𝑥𝑖𝑠−𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑦(𝜃) = +1 +2 ( +𝐽11 + 𝐽22 +(𝐽11 − 𝐽22)𝑒𝑖2𝜃 +(𝐽11 − 𝐽22)𝑒−𝑖2𝜃 +𝐽11 + 𝐽22 +) (B6) + + +Funding. National Key Research and Development Program of China (2022YFA1404800, 2018YFA0306200); +National Natural Science Foundation of China (11922406, 91750202); +Acknowledgments. We thank Mr. Yong Chen for the fruitful discussions and his help provided during this work. +Disclosures. The authors declare no conflicts of interest. +Data availability. Data underlying the results presented in this paper are not publicly available at this time but may +be obtained from the authors upon reasonable request. + +References +1. +H. T. Chen, A. J. Taylor, and N. Yu, “A review of metasurfaces: physics and applications,” Rep. Prog. Phys. +79(7), 076401–076401 (2016). +2. +G. Yoon, T. Tanaka, T. Zentgraf, and J. Rho, “Recent progress on metasurfaces: applications and fabrication,” J. +Phys. D: Appl. Phys. 54(38), 383002 (2021). +3. +J. P. B. Mueller, N. A. Rubin, R. C. Devlin, B. Groever, and F. Capasso, “Metasurface polarization optics: +Independent phase control of arbitrary orthogonal states of polarization,” Phys. Rev. Lett. 118, 113901 (2017). +4. +A. C. Overvig, S. Shrestha, S. C. Malek, M. Lu, A. Stein, C. Zheng, and N. 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Interfaces 14, 33968–33975 (2022). +15. C. Chen, S. Gao, W. Song, H. Li, S. N. Zhu, and T. Li, “Metasurfaces with planar chiral meta-atoms for spin light +manipulation,” Nano Lett. 21, 1815−1821 (2021). +16. H. Wang, B. Zhang, C. Han, and J. Ding, “Polarization-multiplexed wavefront-engineering by all-dielectric +metasurface with asymmetric polarization-decoupled meta-atoms,” Opt. Express 29, 32377–32387 (2021). +17. C. Han, B. Zhang, H. Wang, J. Xu, and J. Ding, “Predicting the eigenstructures of metamaterials with QR-code +meta-atoms by deep learning,” Opt. Lett. 47, 1863–1866 (2022). +18. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, +3rd ed. (Artech House, Norwood, 2005), p. 354. +19. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, New York, 1985), p. 566. +20. R. W. Gerchberg, W. O. Saxton “A practical algorithm for the determination of phase from image and diffraction +plane pictures,” Optik 35, 237−246 (1972). + + + diff --git a/ctAzT4oBgHgl3EQfLvsV/content/tmp_files/load_file.txt b/ctAzT4oBgHgl3EQfLvsV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..691946b2102428c57cd3afe1ab64e26eee6b2be6 --- /dev/null +++ b/ctAzT4oBgHgl3EQfLvsV/content/tmp_files/load_file.txt @@ -0,0 +1,434 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf,len=433 +page_content='Symmetry breaking of Pancharatnam–Berry phase using non-axisymmetric meta-atoms Baifu Zhang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content='1* Yan Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content='1 Zhixing Huang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content='1 Huafeng Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content='1 Ji Xu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content='2 and Jianping Ding3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content='4 1School of Electronic and Optical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Nanjing University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Nanjing 210094,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' China 2College of Electronic and Optical Engineering & College of Flexible Electronics (Future Technology),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Nanjing University of Posts and Telecommunications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Nanjing 210023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' China 3Collaborative Innovation Center of Advanced Microstructures and School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Nanjing University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Nanjing 210093,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' China 4jpding@nju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content='cn zhangbf@njust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content='cn Abstract: The Pancharatnam–Berry (PB) phase in metasurfaces obeys the symmetry restriction, according to which the PB phases of two orthogonal circularly polarized waves are the same but with opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Here, we reveal a general mechanism to break the axisymmetry of meta-atoms in order to break the PB-phase symmetry restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' As a proof of concept, we designed a novel meta-atom with a QR-code structure and successfully demonstrated circular- polarization multiplexing metasurface holography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' This study provides a fundamentally new understanding of the PB phase and opens a path for arbitrary wavefront engineering using asymmetric electromagnetic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Introduction Metasurfaces, as ultrathin electromagnetic (EM) functional layers, have emerged as a comprehensive and compact platform for wavefront engineering in the last decade [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' The overall wavefront engineering by a metasurface results from the linear superposition of EM waves modulated by each meta-atom, which acts as the unit cell of the metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' The conventional phase-modulation mechanisms of a meta-atom include the dynamic phase and Pancharatnam–Berry (PB) phase (or geometric phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' The dynamic phase originates from the accumulated phases of an EM wave propagating in a meta-atom;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' the phase accumulation depends on the refractive index, geometry, and other meta-atom parameters [3,4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' The PB phase arises from the spin–orbit coupling of photons in a meta-atom, as a function of the rotation angle of the anisotropic EM structures (including elliptical structures and rectangular structures) [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' For example, when a circularly polarized EM wave propagates through such structures with a rotation angle θ, the transmitted cross-polarized component adopts the so-called PB phase (Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Conventionally, Φ = ±2θ, where the sign ± is determined by the chirality of the EM wave;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' for instance, the left- and right-circular polarizations (LCP and RCP, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Unlike the dynamic phase, the PB phase is symmetric for both the orthogonal circular polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' In other words, the PB phases of the LCP and RCP have the same absolute value but opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Recently, Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' studied the rotational symmetry of meta-atoms and demonstrated high-order PB phases, which are equivalent to several times the rotation angle (rather than just two times) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' However, the PB phase of these meta-atoms remains symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Therefore, achieving polarization-decoupled EM wavefront engineering using only the geometric phase is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' In recent years, some researchers attempted to break the symmetry of the PB phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' For example, by introducing nonlinear effects, the PB phase of transmitted harmonic waves can be rewritten in the form of ±(n±1) θ, where n is the order of harmonic generation [9–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' However, this method cannot be used to engineer the PB phases of orthogonal circular polarizations arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' In the case of linear processes, Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' proposed to combine the Aharonov– Anandan and PB phases to break the symmetry limitation of the PB phase [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' This proposed method can be applied in EM wave engineering metasurfaces [13,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' proposed a type of planar chiral meta-atom to perform local phase manipulations in metasurfaces and realized a controlled spin decoupling of the orthogonal circularly polarized waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' The studies reported to date trace the origins of asymmetric PB phases from different physical mechanisms and demonstrate polarization-decoupled PB phase modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' However, the identification of a general fundamental factor that can break the symmetry limitation of the PB phase for orthogonal circularly polarized waves is crucial for understanding the relationship between meta-atom structures and their corresponding EM wavefront engineering properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' In this study, we analytically demonstrated that the symmetry restriction of the PB phase originates from the axisymmetry of EM structures for the first time to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' In other words, from the geometrical and topological point of view, a non-axisymmetric meta- atom topology naturally leads to symmetry breaking of the PB phases of two orthogonal circularly polarized waves, although different geometric structures may have different polarization-decoupling mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Further, we analytically established the relationship between the Jones matrices based on the circular- and linear-polarization base vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Moreover, we designed and investigated non-axisymmetric meta-atoms with QR-code structures to verify the breaking of the symmetry restriction of the PB phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' As a proof of concept, we designed a circularly polarized multiplexed hologram using a metasurface consisting of QR-code meta-atoms and numerically demonstrated our proposed theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' The results of this study provide a fundamentally new understanding of the PB phase and light– matter interactions in nanophotonics and can thus promote more advanced metasurface- and metamaterial-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Theory of PB phase for non-axisymmetric meta-atoms First, we derive a general form of the PB phase for a meta-atom with an arbitrary topology, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=', a non-axisymmetric structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Note that the structure of a three-dimensional (3D) meta-atom is also called a “non-mirror symmetric” structure instead of a non-axisymmetric structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' However, considering that the topologies of meta-atoms usually vary in the top-view plane and remain unchanged in the height dimension, we perform the analysis using the “non- axisymmetry” concept for simplicity but without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' For an EM wave incident on a meta-atom, the relationship between the transmitted and incident waves can be established using the Jones matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Usually, we use linearly polarized base vectors to describe the incident and transmitted EM waves as follows: 𝐸𝑖𝑛 = 𝑎1𝑥̂ + 𝑎2𝑦̂ = (𝑎1 𝑎2), (1) 𝐸𝑜𝑢𝑡 = 𝐽𝐸𝑖𝑛 = (𝐽11 𝐽12 𝐽21 𝐽22) (𝑎1 𝑎2), (2) where Ein and Eout are the incident and transmitted EM waves, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' 𝑥̂ and 𝑦̂ are the base vectors of the x and y linear polarizations, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' a1 and a2 are the corresponding complex amplitudes with 𝑎1 2 + 𝑎2 2 = 1 as the normalization condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' and J is the Jones matrix for linear-polarization base vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' For an arbitrary meta-atom structure, the four elements of J are all non-zero values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' However, for an axisymmetric structure, J21 = J12 = 0 (details in the Appendix), which suggests that a linear-polarization-decoupled metasurface can be designed using axisymmetric meta-atoms as discussed in our previous work [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Introducing circular-polarization base vectors to the above equations, we obtain 𝐸𝑖𝑛 = 𝑏1𝐿̂ + 𝑏2𝑅̂ = (𝑏1 𝑏2), (3) 𝐸𝑜𝑢𝑡 = 𝑆𝐸𝑖𝑛 = (𝑆11 𝑆12 𝑆21 𝑆22) (𝑏1 𝑏2), (4) where 𝐿̂ and 𝑅̂ are the base vectors of LCP and RCP, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' b1 and b2 are the corresponding complex amplitudes with 𝑏1 2 + 𝑏2 2 = 1 as the normalization condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' and S is the Jones matrix for circular-polarization base vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' If we further consider the relationship between two sets of orthogonal base vectors as 𝐿̂ = √2 2 (𝑥̂ − 𝑖𝑦̂) and 𝑅̂ = √2 2 (𝑥̂ + 𝑖𝑦̂), and conduct some analytical derivations (details in the Appendix), then we can rewrite the S matrix in the form of J matrix as follows: { 𝑆11 = 1 2 [(𝐽11 + 𝐽22) − 𝑖(𝐽12 − 𝐽21)] 𝑆21 = 1 2 [(𝐽11 − 𝐽22) − 𝑖(𝐽12 + 𝐽21)] 𝑆12 = 1 2 [(𝐽11 − 𝐽22) + 𝑖(𝐽12 + 𝐽21)] 𝑆22 = 1 2 [(𝐽11 + 𝐽22) + 𝑖(𝐽12 − 𝐽21)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' (5) Further, if the meta-atom rotates by an angle θ, then the Jones matrix is expressed as 𝐽(𝜃) = 𝑅(−𝜃)𝐽𝑅(𝜃) = (𝑐𝑜𝑠𝜃 −𝑠𝑖𝑛𝜃 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃 ) (𝐽11 𝐽12 𝐽21 𝐽22) ( 𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃 −𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' (6) We substitute Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' (2) and (6) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' (5), and finally obtain the most general form of the PB phase for an arbitrary meta-atom using circular-polarization base vectors as follows: { 𝑆11(𝜃) = 1 2 [(𝐽11 + 𝐽22) − 𝑖(𝐽12 − 𝐽21)] 𝑆21(𝜃) = 1 2 [(𝐽11 − 𝐽22) − 𝑖(𝐽12 + 𝐽21)]𝑒−𝑖2𝜃 𝑆12(𝜃) = 1 2 [(𝐽11 − 𝐽22) + 𝑖(𝐽12 + 𝐽21)]𝑒𝑖2𝜃 𝑆22(𝜃) = 1 2 [(𝐽11 + 𝐽22) + 𝑖(𝐽12 − 𝐽21)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' (7) Compared to the linear Jones matrix J(θ), S(θ) intuitively exhibits numerous important properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' First, the diagonal elements S11 and S22 remain unchanged even if the meta-atom is rotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Second, the anti-diagonal elements S21 and S12 for the transmitted cross-polarizations are naturally symmetry-broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' In other words, the cross-polarization transmissions of the incident circularly polarized EM waves are inherently asymmetric, which is contrary to the traditional view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' PB phases are subjected to symmetry restrictions in conventional meta-atoms such as those with rectangular, elliptical, and other structures, which have been widely studied in previous works, because for the aforementioned axisymmetric topologies, J12 = J21 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Thus, the asymmetric elements S21(θ) and S12(θ) degenerate into the most known forms with a symmetric PB phase: { 𝑆21(𝜃) = 1 2 (𝐽11 − 𝐽22)𝑒−𝑖2𝜃 𝑆12(𝜃) = 1 2 (𝐽11 − 𝐽22)𝑒𝑖2𝜃 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' (8) The conventional symmetry restriction of the PB phase reportedly originates from the axisymmetry of meta-atoms, and thus breaking the axisymmetry of the meta-atoms intrinsically leads to breaking of the PB-phase symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Notably, instead of the structural symmetry, the optical mode (electromagnetic field) inside the meta-atom structure should be adopted as a more general physical quantity in this case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' however, these two factors are usually consistent with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Thus, for simplicity, we perform our analysis from the structural point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Next, we rewrite S21(θ) and S12(θ) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' (7) in a more intuitive form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' If S21 and S12 simplify to 𝐴21𝑒𝑖𝜑21 and 𝐴12𝑒𝑖𝜑12, respectively, then S21(θ) and S12(θ) can be expressed as: {𝑆21(𝜃) = (𝐴21𝑒𝑖𝜑12)𝑒−𝑖(2𝜃−∆𝜑) 𝑆12(𝜃) = (𝐴12𝑒𝑖𝜑12)𝑒𝑖2𝜃 , (9) where A12 and A21 are the amplitudes of S21 and S12, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' 𝜑21 and 𝜑12 are the argument angles of S21 and S12, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' and ∆𝜑 = 𝜑21 − 𝜑12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' The symmetry of PB phase is broken by the key phase ∆𝜑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Schematic of the non-axisymmetric meta-atom with QR-code structure In order to break the symmetry restriction of the PB phase, the meta-atom should break the axisymmetry in the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Here, we design a novel dielectric meta-atom with a QR-code structure to eliminate any geometric symmetry as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' This type of QR-code topology, which was investigated in our previous study on perfect metamaterial absorbers, possesses a large number of degrees of freedom and can thus provide rich properties for wavefront engineering [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' In this study, the meta-atom, with a period P of 800 nm, consists of a silica substrate and arrays of square silicon pillars that appear as a QR code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Each QR code consists of 5 × 5 pixels, and each pixel is randomly set as either a square silicon pillar (length: 100 nm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' height: 1400 nm) or air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Because the generated QR-code topology is completely random, it easily and thoroughly breaks the axisymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' 1 Schematic of the non-axisymmetric meta-atom with QR-code structure: (a) 3D view and (b) top view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' The parameters are set as P = 800 nm, L = W = 100 nm, and H = 1400 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' We conducted full-wave 3D finite-difference time-domain (FDTD) simulations to design and investigate the QR-code meta-atom structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' In the simulations, we set the refractive index of the silica substrate as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content='45, and derived the dielectric constant values of silicon by fitting Palik’s experiment data to the Drude–Lorentzian model [18,19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' For the meta-atom simulation, we used the period boundaries in the horizontal direction and perfectly matched layers (PMLs) in the longitudinal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' By randomly setting each pixel of the QR-code structure, we obtained a variety of non-axisymmetric meta-atoms and numerically calculated their parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' All the simulated meta-atoms show asymmetric PB phases, which are discussed in detail in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Breaking the symmetry restriction of PB phase using non-axisymmetric meta-atoms We designed and numerically evaluated 13,000 meta-atom structures with randomly arrayed 5×5 QR-code nanopillars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' All the QR-code meta-atoms show decoupled PB phases for LCP and RCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' We selected structures with transmission coefficients higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content='7 to ensure efficient polarization-multiplexing applications, and the PB phases of these selected structures are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Evidently, the PB phases of the LCP and RCP cover the 0–2π range and (a) (b) Si SiOz Pbreak the symmetry restriction, consistent with the predictions of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' (7) and (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' For simplicity but without loss of generality, we do not rotate the meta-atoms in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' 2, indicating that the rotation angle θ is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' The corresponding results suggest that the polarization- decoupled PB phases, which originate from non-axisymmetric structures, are new degrees of freedom for engineering EM wavefronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' In addition, we can independently engineer the PB phases for orthogonal circular polarizations without rotating the meta-atoms in the full range from 0 to 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' 2 PB phases of different meta-atom structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' (a) PB phases of the QR-code structures shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' (b) PB phases of different axisymmetric structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' We investigated a variety of axisymmetric meta-atom structures as a control group, and their PB phases are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' All axisymmetric meta-atoms have the same period (800 nm) and height (1400 nm) as the non-axisymmetric ones, but different topologies in the top view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' All PB phases are located on the y = x line, irrespective of the meta-atom topology (such as rectangle, ellipse, trapezoid, cross, T-shape, H-shape, or hollow rectangle), implying that all PB phases of the two orthogonal circular polarizations have the same value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' This is the symmetry restriction of conventional PB phases, which is predicted in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' (8), and originates from axisymmetric structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' 2, we can verify our analytical predictions of the PB phases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' that is, decoupling of the PB phases can be achieved under a broken axial symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Furthermore, the decoupled PB phase, independent of the rotation angle of the meta-atom, provides a new degree of freedom in wavefront engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' As a proof of concept, we designed a circular-polarization multiplexing metasurface hologram based on our proposed QR-code meta-atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' The design of the circular-polarization-decoupled metasurface hologram is illustrated in a flowchart in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' First, we calculated the phase-only computer-generated holograms (CGHs) for left- and right-circular polarized incident light beams using the Gerchberg–Saxton (GS) algorithm [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' We coded two different images for the two orthogonal polarizations and obtained two output holographic phase diagrams after the GS iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Second, we performed an eight-level phase quantization to rewrite the two phase diagrams into two phase matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' In this process, we selected 8 × 8 meta-atom structures from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' 2(a) to establish the eight-level phase library, which was then searched to identify the meta-atom structures that fulfilled both the phase requirements for the LCP and RCP holograms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' to identify these structures, a one-by- one element search of the phase matrices was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Finally, we simulated the desired metasurface hologram with 64 types of QR-code structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Due to the limited computing resources, we could simulate a metasurface with 101 × 101 periods as a proof of concept, which successfully demonstrated our theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' As the period was set to 800 nm, the metasurface hologram could cover an area of 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content='8 × 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content='8 μm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Phase of S21 (unit: π) Phase of S12 (unit: π) (a) Phase of S21 (unit: π) Phase of S12 (unit: π) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' 3 Design flowchart of the circular-polarization-decoupled metasurface hologram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Circular-polarization-decoupled metasurface holography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' (a) and (e) are the original images for LCP and RCP illuminations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' (b) and (f) correspond to the phase distributions of the calculated holograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' (c) and (g) are the images of the theoretical reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' (d) and (h) correspond to the reconstructed images of the simulated metasurface hologram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' The designed metasurface hologram was simulated using 3D FDTD and is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' 3, and the reconstructed images for LCP and RCP are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Figures 4 (a) and (e) show the 101 × 101-pixel dog and cat images as the original images, which are illuminated by left- and right-circularly polarized light, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Thus, the images reconstruct using the PB phases incorporate the cross-polarization components RCP and LCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' The phase-only CGHs of the two orthogonal polarizations, simulated using the GS algorithm, are show in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' 4(b) and (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' As the control group, the images of the theoretical reconstructions are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' 4(c) and (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' The reconstructed images of the simulated metasurface hologram, which was illuminated by different circularly polarized EM waves, are plotted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' 4(d) and (h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' We can see that the reconstructed images of the metasurface hologram show good agreement with the original and theoretically reconstructed images, thereby confirming the feasibility of decoupling circularly polarized light via wavefront engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' This result demonstrates that non-symmetric meta-atom structures can indeed break the symmetry restriction of PB phases and can provide new insights into metasurfaces and metamaterials to promote their application in various fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Conclusion In conclusion, we theoretically established a general formula to describe the wavefront engineering property of a meta-atom using the Jones matrix and circular-polarization base vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' The analytical result indicates that the non-axisymmetry of meta-atom leads to the breaking of the symmetry restriction of the PB phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' As an illustrative example, we designed π 0 π 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content='5 0 π 0 π 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content='5 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content='5 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content='5 0 (a) (e) (b) (f) (c) (g) (d) (h) Design of CGH 8-level-phase library Label Plharse Designed metasurface 心 LCP 1/8-2x GS I Image 8Z Iteration 3/8 -2x E 4/8 -2元 RCP 5/8 -2元 Image .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content='5 6/8-2# 7/8-2元a novel QR-code meta-atom to break the topological axisymmetry and realize polarization- decoupled PB-phase engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Further, we designed a circularly polarized multiplexed metasurface hologram using our proposed QR-code meta-atoms to reveal the potential of symmetry-broken PB phases in various applications in metasurfaces and metamaterials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' We believe that this study will expand the understanding of the PB phase in a fundamental way and will facilitate the development of design methodologies for EM structures, which can be used for arbitrary wavefront engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Appendix A: Derivation of Jones matrix S When a left-circularly polarized light beam is incident on an arbitrary meta-atom,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' the transmitted (or reflected) light is 𝐸𝑜𝑢𝑡 = 𝑆𝐸𝑖𝑛 = (𝑆11 𝑆12 𝑆21 𝑆22) (1 0) = 𝑆11𝐿̂ + 𝑆21𝑅̂ = 𝑆11 √2 2 (𝑥̂ − 𝑖𝑦̂) + 𝑆21 √2 2 (𝑥̂ + 𝑖𝑦̂) (A1) Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' when a right-circularly polarized light beam incidents,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' the output light field is expressed as: 𝐸𝑜𝑢𝑡 = 𝑆𝐸𝑖𝑛 = (𝑆11 𝑆12 𝑆21 𝑆22) (0 1) = 𝑆12𝐿̂ + 𝑆22𝑅̂ = 𝑆12 √2 2 (𝑥̂ − 𝑖𝑦̂) + 𝑆22 √2 2 (𝑥̂ + 𝑖𝑦̂) (A2) We can rewrite the above equations using linear-polarization base vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' For a left-circularly polarized incident light beam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' the transmitted electric field is given by 𝐸𝑜𝑢𝑡 = 𝐽𝐸𝑖𝑛 = (𝐽11 𝐽12 𝐽21 𝐽22) ( √2 2 − √2 2 𝑖 ) = √2 2 (𝐽11 − 𝑖𝐽12)𝑥̂ + √2 2 (𝐽21 − 𝑖𝐽22)𝑦̂ (A3) Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' for a right-circularly polarized incident light beam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' we obtain 𝐸𝑜𝑢𝑡 = 𝐽𝐸𝑖𝑛 = (𝐽11 𝐽12 𝐽21 𝐽22) ( √2 2 √2 2 𝑖 ) = √2 2 (𝐽11 + 𝑖𝐽12)𝑥̂ + √2 2 (𝐽21 + 𝑖𝐽22)𝑦̂ (A4) The physical processes represented by the aforementioned equations remain identical irrespective of using a circular- or linear-polarization base vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Thus, we solve Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' (A1)– A(4) simultaneously and establish the analytical relationship between J and S as follows: { 𝑆11 = 1 2 [(𝐽11 + 𝐽22) − 𝑖(𝐽12 − 𝐽21)] 𝑆21 = 1 2 [(𝐽11 − 𝐽22) − 𝑖(𝐽12 + 𝐽21)] 𝑆12 = 1 2 [(𝐽11 − 𝐽22) + 𝑖(𝐽12 + 𝐽21)] 𝑆22 = 1 2 [(𝐽11 + 𝐽22) + 𝑖(𝐽12 − 𝐽21)] (A5) Appendix B: Axisymmetry of meta-atoms For an x-polarized incident light beam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' the output light wave is given by (𝐽11 𝐽12 𝐽21 𝐽22) (1 0) = 𝐽11𝑥̂ + 𝐽21𝑦̂ (B1) If a mirror operation is performed on the meta-atom along the symmetric axis (for example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' the x-axis),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' then the Jones matrix 𝐽′ becomes (𝐽11 ′ 𝐽12 ′ 𝐽21 ′ 𝐽22 ′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' and the output becomes (𝐽11 ′ 𝐽12 ′ 𝐽21 ′ 𝐽22 ′ ) (1 0) = 𝐽11 ′ 𝑥̂ + 𝐽21 ′ 𝑦̂ (B2) According to the property of mirror operation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' the y-polarization component effectively undergoes a phase retardation of e𝑖𝜋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' and thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' we obtain 𝐽11𝑥̂ = 𝐽11 ′ 𝑥̂,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' 𝐽21𝑦̂ = 𝐽21 ′ 𝑦̂ ∙ 𝑒𝑖𝜋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' which implies 𝐽11 ′ = 𝐽11，' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content='𝐽21 ′ = −𝐽21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Similarly, in the case of the y-polarization incident light beam, we can obtain 𝐽12 ′ = −𝐽12， 𝐽22 ′ = 𝐽22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Thus, the relationship between J and 𝐽′ can be expressed as 𝐽′ = (𝐽11 ′ 𝐽12 ′ 𝐽21 ′ 𝐽22 ′ ) = ( 𝐽11 −𝐽12 −𝐽21 𝐽22 ) (B3) For an axisymmetric meta-atom, mirror operation on its symmetric axis does not change its geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Thus, we obtain following equation: 𝐽′ = (𝐽11 ′ 𝐽12 ′ 𝐽21 ′ 𝐽22 ′ ) = ( 𝐽11 −𝐽12 −𝐽21 𝐽22 ) = (𝐽11 𝐽12 𝐽21 𝐽22) = 𝐽 (B4) which indicates 𝐽12 = −𝐽12 = 0，𝐽21 = −𝐽21 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Thus, the Jones matrix J of the axisymmetric meta-atoms is simplified to 𝐽𝑎𝑥𝑖𝑠−𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑦 = (𝐽11 0 0 𝐽22) (B5) Further, if the axisymmetric meta-atom rotates by an angle θ, then the corresponding Jones matrix S(θ) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' (7) is simplified to 𝑆𝑎𝑥𝑖𝑠−𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑦(𝜃) = 1 2 ( 𝐽11 + 𝐽22 (𝐽11 − 𝐽22)𝑒𝑖2𝜃 (𝐽11 − 𝐽22)𝑒−𝑖2𝜃 𝐽11 + 𝐽22 ) (B6) Funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' National Key Research and Development Program of China (2022YFA1404800, 2018YFA0306200);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' National Natural Science Foundation of China (11922406, 91750202);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' We thank Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Yong Chen for the fruitful discussions and his help provided during this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' Disclosures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfLvsV/content/2301.01118v1.pdf'} +page_content=' The authors declare no conflicts of interest.' metadata={'source': 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Thingna1, 2, 7, † +1Center for Theoretical Physics of Complex Systems, +Institute for Basic Science (IBS), Daejeon 34126, Republic of Korea. +2Basic Science Program, Korea University of Science and Technology, Daejeon 34113, Republic of Korea. +3School of Physical and Mathematical Science, Nanyang Technological University, Singapore 639798, Singapore +4Department of Physics, Indian Institute of Technology-Bombay, Powai, Mumbai 400076, India. +5Centre of Excellence in Quantum Information, Computation, Science and Technology, +Indian Institute of Technology Bombay, Powai, Mumbai 400076, India. +6Centre for Quantum Technologies, National University of Singapore, +3 Science Drive 2, Singapore 117543, Singapore. +7Department of Physics and Applied Physics, University of Massachusetts, Lowell, MA 01854, USA. +(Dated: January 12, 2023) +Synchronization between limit cycle oscillators can arise through entrainment to an external drive +or through mutual coupling. The interplay between the two mechanisms has been studied in clas- +sical synchronizing systems, but not in quantum systems. Here, we point out that competition and +cooperation between the two mechanisms can occur due to phase pulling and phase repulsion in +quantum systems. We study their interplay in collectively driven degenerate quantum thermal ma- +chines and show that these mechanisms either cooperate or compete depending on the working mode +of the machine (refrigerator or engine). The entrainment-mutual synchronization interplay persists +with an increase in the number of degenerate levels, while in the thermodynamic limit of degener- +acy, mutual synchronization dominates. Overall, our work investigates the effect of degeneracy and +multilevel scaling of quantum synchronization and shows how different synchronizing mechanisms +can cooperate and compete in quantum systems. +Introduction.– Synchronization is a ubiquitous phe- +nomenon in which stable phase relations emerge between +multiple limit cycle oscillators [1]. There are two main +mechanisms that give rise to synchronization: i. +En- +trainment that refers to synchronization of an oscillator +by unidirectional coupling to a periodic external drive +[2], and ii. Mutual synchronization which refers to the +adjustment of rhythms of two or more mutually coupled +oscillators, such as in the widely-known Kuramoto model +[3]. These two mechanisms may coexist in some systems +[4–7] and their interplay has also been experimentally +studied in globally coupled electrochemical oscillators [8]. +In the same spirit as classical synchronization, quan- +tum synchronization is often studied through entrain- +ment [9–14] or mutual coupling [15–21] and has been +experimentally observed recently [22–24]. However, un- +like classical synchronization, the coexistence and the in- +terplay between these two mechanisms in the quantum +regime has not been investigated. Understanding this in- +terplay is crucial in the control of various quantum tech- +nologies where both driving and interaction are impor- +tant such as in superradiant lasers [16], coupled time- +crystals [25], and coupled heat engines [26–29]. +In this work, we show that the phases of steady-state +coherences follow a phase synchronization model, where +the external entraining drive competes with the mu- +tually coupled phases. +This opens up the possibility +of observing well-studied classical phenomena, such as +synchronization-anti-synchronization transition [30] and +chimera [31, 32], in the quantum regime. Our framework +applies to generic quantum systems, with one or more ex- +ternal drives that couple the coherences that themselves +are mutually coupled, either coherently or dissipatively, +leading to an interplay between entrainment and mutual +synchronization. +As a concrete example, we consider a degenerate mul- +tilevel generalization of the Scovil–Schulz-DuBois maser +heat engine [33], where the external collective drive con- +nects transitions between the degenerate manifold and +the first-excited state [34]. The states within the degen- +erate manifold mutually interact to form a stable +col- +lective symmetric (in-phase) and anti-symmetric (out-of- +phase) superposition (mutual synchronization). At the +same time, the external drive causes the phases within +the degenerate manifold to be aligned in-phase with the +drive (entrainment). +In the engine regime, stimulated +emission consumes the collective symmetric superposi- +tion state thereby enhancing the population of the anti- +symmetric state. Thus, there is competition between en- +trainment (in-phase) and mutual synchronization (out- +of-phase). In the refrigerator regime, the stimulated ab- +sorption enhances the population of the collective sym- +metric superposition state thereby always cooperating +with entrainment. Our work sheds light on the synergis- +tic interplay between entrainment and mutual synchro- +nization in quantum systems. +Quantum synchronization in D-level systems.– Quan- +tum synchronization has been studied in systems with +continuous degrees of freedom such as oscillators [9– +11, 13, 15, 17, 35] and discrete degrees of freedom such +as spin-1 systems [12, 14, 20]. A wide variety of mea- +sures, based on various physical and mathematical mo- +arXiv:2301.04322v1 [quant-ph] 11 Jan 2023 + +2 +tivations such as phase-space based measures [9, 12, 20], +correlation measures [36], and information-theoretic mea- +sures [37] has been used to quantify synchronization. +In this work, we use the phase-space based measure +built on the Husimi-Q phase space representation [38, 39] +of the steady-state ρss with respect to SU(D) coherent +state [39, 40] defined as +Q[ρss] = +D! +πD−1 ⟨αD|ρss|αD⟩ , +(1) +where |αD⟩ = �D +n=1 αn |n⟩ is the SU(D) coherent state +with coefficients +αn = +� +eiφn cos θn +�n−1 +k=1 sin θk +1 ≤ n < D +eiφD �D−1 +k=1 sin θk +n = D, +(2) +where it is implicitly assumed that the product term is +identity for n = 1 and the reference phase φ1 = 0. The +synchronization measure is given by the difference be- +tween integrating out the angles θk corresponding to the +population degrees of freedom and doing the same for the +uniform measure, given by +S(φ1, · · · , φD−1) = +� +Q[ρss]dΘ − +1 +(2π)D−1 += +1 +2D+1πD−2 +� +n̸=m +ρss +nmei(φm−φn), (3) +which lives on a D − 1 dimensional torus (see Append. +A). The distribution S(φ1, · · · , φD−1) is zero everywhere +for a diagonal steady-state which is interpreted as a limit +cycle [37] possessing stable amplitudes (fix diagonal ele- +ments) but free phases. The notion of free-phase in a such +diagonal limit cycle is analogous to a classical stochastic +limit cycle whose phase distribution approaches a uni- +form distribution in the steady-state [1, 13, 14, 41, 42]. +We associate the peak of S(φ1, · · · , φD−1) as a phase- +space synchronization measure [12, 20, 42], +Smax = +max +φ1,··· ,φD−1 +1 +2D+1πD−2 +� +n̸=m +ρss +nmei(φm−φn). +(4) +The synchronization measure, Smax only depends on +the steady-state coherences. +However, we note that a +high value of Smax requires all phase preferences Φij = +arg(ρss +ij ) to be compatible, i.e., Φij − Φjk = Φik ∀i ̸= j ̸= +k, a condition that is stronger than the mere presence of +coherences. +Degenerate thermal maser.– Entrainment in quantum +systems is the result of an interplay between coherent +driving and dissipation [10, 12]. The system we consider +is depicted in Fig. +and consists of (N + 2) levels whose +bare Hamiltonian is given by, +H0 = ω1 |1⟩ ⟨1| + +N+1 +� +j=2 +ωj |j⟩ ⟨j| , +(5) +|0⟩ +|1⟩ +|2⟩ +... +|N + 1⟩ +Th +Tc +λ, Ω +∆ +ω2 +ω1 +FIG. 1. +Schematic of the degenerate quantum thermal +maser, which is a generalization of the standard Scovil– +Schulz-DuBois three-level thermal maser [33]. Here, N is the +number of states in the degenerate manifold and here we focus +on the case ∆ = 0. The near-degenerate case where ∆ ̸= 0 is +discussed in the accompanying manuscript [43] +with ωj+1 > ωj, ω0 = 0. The upper N levels are degen- +erate with ω2 = ω3 = · · · = ωN+1. Although we work in +the limit of exact degeneracy, our main results hold even +in the near-degenerate scenario and will be considered in +detail in an accompanying Ref. [43]. +This system is driven by a monochromatic drive V (t) +of frequency Ω given by +V (t) = +N+1 +� +j=2 +λjeiΩt |1⟩ ⟨j| + h.c. +(6) +This drive can be rewritten as a coupling to a collective- +transition mode |1⟩ ↔ |J⟩ = (1/λeff) � +j λj |j⟩ with +λeff = +�� +j |λj|2 being the effective coupling strength. +Such collective drives are realizable in an ensemble of +atoms driven by light, if the inter-atomic distance is much +smaller than the wavelength of the light, such as in the +case of Dicke superradiance [44]. +The system is acted upon by a dissipator +D[ρ] = +2 +� +µ=1 +� +ΓcµL[cµ]ρ + +N+1 +� +j=2 +ΓhµL[hj +µ]ρ +� +, +(7) +which leads to a multilevel generalization of the Scovil– +Schulz-DuBois maser heat engine [33, 34]. The dissipator +L[X]ρ = 2XρX†−{X†X, ρ} is of the Lindblad form such +that the hot (cold) bath with jump operators hj +1 = hj† +2 = +|0⟩ ⟨j| (c1 = c† +2 = |0⟩ ⟨1|) induce transitions between the +ground state and the degenerated manifold (first-excited +state). The associated rates follow local-detailed balance +and are given by Γh1(c1) = γh(c)(1 + nh(c)) and Γh2(c2) = +γh(c)nh(c) with γh(c) being the effective system-bath cou- +pling strength and nh(c) = +� +exp(βh(c)ω2(1)) − 1 +�−1 be- +ing the Bose-Einstein distribution at inverse temperature + +3 +βh(c). The action of the heat baths leads to a population +inverted steady state between the first-excited state |1⟩ +and the degenerated manifold {|j⟩, ∀j = 2, · · · , N + 1} +if nh > nc. If there is population inversion, the system +behaves as a maser heat engine [45]. However, if nh < nc, +population inversion is lost and the system behaves as a +refrigerator by attenuating the drive [45]. We can rewrite +the Hamiltonian in a frame co-rotating with the drive as +˜H = (Ω/2)(�N+1 +j=2 |j⟩ ⟨j| − |1⟩ ⟨1|) giving us the rotating +frame quantum master equation, +d˜ρ +dt = −i[H0 − ˜H + ˜V , ˜ρ] + D[˜ρ], +(8) +where ˜O ≡ e−i ˜ +HtOei ˜ +Ht (O = ρ, V ) is an operator in the +rotated frame with ˜V = �N+1 +j=2 λj |1⟩ ⟨j| + h.c.. +Competition vs cooperation.– Equation (8) can be solved +analytically for the case of homogeneous driving strength +λj = λ (∀j = 2, · · · , N + 1) and resonant driving Ω = +ω2 − ω1. +In this case, the steady-state coherences are +given by +˜ρss +1j = iλ(nc − nh)γcγh(1 + nh) +F(N, nh, nc, γc, γh, λ) , +(9) +˜ρss +jl = +λ2γc(nc − nh) +F(N, nh, nc, γh, γc, λ), +(10) +where j, l = 2, · · · , N + 1, j ̸= l and the function +F(N, nh, nc, γc, γh, λ) = AN 2 + BN + C with A, B, and +C being positive constants that depend on all remaining +parameters (see Append. B for the explicit expressions +for these constants). +The non-degenerate coherences (˜ρ1j) are directly in- +duced (i.e., ∝ λ) by the drive whereas the degenerate co- +herences (˜ρjl) are an indirect consequence (∝ λ2) of the +collective nature of the drive. Their differences are clear +as one transforms back to the original frame in which +ρ1j = ˜ρ1je−iΩt and ρjl = ˜ρjl. +The phase preferences +induced by ρ1j rotate with the driving frequency while +that of ρjl remain stationary in the original frame. Both +of these coherences affect the phase distributions of the +states within the degenerate manifold. For these reasons, +we infer that there are two synchronization mechanisms +at play in this system, entrainment induced directly by +the drive and mutual coupling that occurs due to the +presence of a degenerate manifold. Entrainment induces +phases relative to driving whose effect is the emergence +of stable non-degenerate coherences ˜ρss +1j. On the other +hand, mutual coupling induces a relative phase between +states in the degenerated manifold independent of the +driving phase, which is reflected by stable degenerate co- +herences ˜ρss +jl . +Recall that we have denoted Φij = arg(˜ρss +ij ) as the +steady-state phase preferences. When there are multi- +ple of such preferences, synchronization requires all the +phase relations to be compatible, i.e. Φij − Φjk = Φik +(i ̸= j ̸= k). However, we find that in our system such a +a +−π +− π +2 +0 +π +2 +π +−π +− π +2 +0 +π +2 +π +ϕ31 +ϕ21 +−5 · 10−4 +5 · 10−4 +S(ϕ21, ϕ31) +b +−π +− π +2 +0 +π +2 +π +−π +− π +2 +0 +π +2 +π +ϕ31 +ϕ21 +0.5 +1 +1.5 +2 +0 +1 +2 +c +nh/nc +Smax × 10−4 +−1 −0.5 +0 +0.5 +1 +1 +2 +3 +d +λ2/λ3 +Smax × 10−4 +FIG. 2. Interplay between entrainment and mutual coupling +for N = 2. Panels a and b show phase quasi-distribution func- +tion S(ϕ21, ϕ31) [Eq. (3)] where ϕij = φi − φj in the engine +regime (nh/nc = 100). For k = 3, S(ϕ21, ϕ31) shows a local- +ized maximum when the phases are in-phase (ϕ21 − ϕ31 ≈ 0 +in the red-region in a, entrainment-dominant). Whereas for +k = 0.75, when S(ϕ21, ϕ31) is maximized the phases do not +localize but their difference is out-of-phase (ϕ21 − ϕ31 ≈ π +in the red-region in b, mutual coupling dominant). Panel c +shows Smax (solid circle) as a function of nh/nc with the +solid line representing the analytic prediction of Eq. (11). +The dashed line is the entrainment contribution to Smax, +i.e., (|ρ12| + |ρ13|)/16π2. The vertical dotted line represents +the boundary between refrigerator (nh/nc < 1) and engine +(nh/nc > 1) regimes. Panel d shows Smax (solid circle) and +(|ρ12| + |ρ13|)/16π2 (dashed line) plotted against inhomoge- +neous driving strength ratio |λ2/λ3| ≤ 1 in the engine (red) +and refrigerator (blue) regimes indicating competition (coop- +eration) between entrainment and mutual coupling is robust +in the engine (refrigerator) regime. The other parameter val- +ues are ω2 = ω3 = 3ω1, Ω = ω2 − ω1, γc = 0.2ω1, γh = +0.05ω1, nc = 0.5, and λ2 = 0.1ω1 +condition is only satisfied in the refrigerator regime where +Φ1j = π/2 (∀j) and Φjl = 0 (j ̸= l). In the engine regime, +we have Φ1j = −π/2 (∀j) and yet Φjl = π (j ̸= l). We +interpret this as a result of an interplay between entrain- +ment and mutual coupling. +We find that entrainment +always pulls the degenerate states to be in-phase (Fig. +2a). Mutual coupling prefers out-of-phase configuration +in the engine regime (Fig. 2b), and in-phase configura- +tion in the refrigerator regime. Consequently, we expect +entrainment and mutual coupling to cooperate in the re- +frigerator regime and compete in the engine regime. +The competition and cooperation are obvious when we +calculate the phase space synchronization measure Smax +[see Eq. (4)]. In general, this requires optimization over + +4 +2 +8 +14 +20 +2 +4 +×10−2 +a +N +(2π)NSmax +2 +8 +14 +20 +5 +10 +15 +20 +×10−2 +b +N +(2π)NSmax +0 +π +3π +2 +π +2 +c +0 +π +3π +2 +π +2 +d +FIG. 3. +Panels a-b show Smax = (2π)NSmax (solid cir- +cle) compared with its entrainment contribution 1 +4 +�N+1 +j=2 |ρ1j| +(empty circle) as a function of N. +The error bar is calcu- +lated from 102 random realizations of driving strength ratio +λj/λ2 ≤ 1 for λj ≥ 0 and j = 3, · · · , N + 1 with λ2 held +constant. The solid lines are the curve fits using Smax ∝ N α +with α = −0.72 (a) and = 0.69 (b). Panels c-d show optimum +phases {ϕopt +j1 } for N = 20 in in the engine (c, nh/nc = 10) +and refrigerator (d, nh/nc = 0.4) regimes plotted on a unit +circle. The different opacity represents different realizations +of λj/λ2 (j = 3, · · · , N + 1). All the phases in all realizations +coalesce to a single data point in the refrigerator case. All +other parameters are the same as Fig. 2. +N variables which we calculate analytically for N = 2 +(see Append. C) +Smax = +1 +16π2 × +� +� +� +� +� +� +� +� +� +|˜ρss +12| + |˜ρss +13| + |˜ρss +23| +if nhnc & k>2 +� +1 + k2 +2 +� +|˜ρss +23| +if nh>nc & k<2, +(11) +where k = γh(1 + nh)/λ = |˜ρss +12|/|˜ρss +23| = |˜ρss +13|/|˜ρss +23| is the +dissipation-to-driving ratio. The set of optimal phases +(ϕopt +21 , ϕopt +31 ) ≡ (ϕ21, ϕ31)|S=Smax evaluated in Append. C +are given by, +(ϕopt +21 , ϕopt +31 ) = +� +� +� +� +� +� +� +� +−π +2 , −π +2 +� +if nhnc & k>2 +(χ, π − χ) & (π − χ, χ) +if nh>nc & k<2, +(12) +where ϕij = φi − φj and χ = arcsin(k/2). Equations +(11)-(12) show the effect of the coherent drive and bath +couplings on the synchronous dynamics of the system. +Cooperation in the refrigerator regime (nc > nh) is re- +flected by the fact that each component of the magni- +tude of coherence adds up in the synchronization mea- +sure Smax, whereas in the engine case there is competi- +tion since the mutual coupling component |ρss +23| reduces +the effect of the entrainment contribution |ρss +12|+|ρss +13|. In +other words, the phases are either equal in some cases or +they are arranged antipodally in other cases, as shown in +Eq.(12). +In the engine regime, Smax is also divided into regimes +where entrainment is dominant (k > 2) and where the +mutual coupling is dominant (k < 2). For the entrain- +ment dominant regime, the competition is apparent from +the negative contribution of |ρss +23| to Smax. +Note that +this is different from the previously reported phenomenon +of synchronization blockade [14, 46], in our case, Smax +can not vanish except for λ = 0 or nh = nc where the +steady-state is diagonal (see Append. D). The transition +from entrainment to mutual coupling dominant regime +is shown in Figs. 2a-b where we plot the phase distribu- +tion S(ϕ21, ϕ31) for different k values. In particular, we +see that as we cross k = 2, the relative phases go from +in-phase to out-of-phase. Moreover, the localization pat- +tern changes from a point localization to ring localization +(on a torus), wherein the latter only the relative phase +ϕ23 = ϕ21 − ϕ31 is fixed, indicating that entrainment is +lost. +The competition and cooperation observed is also ro- +bust with respect to all values of individual driving +strength ratio λ2/λ3 as shown in Fig. 2d. +Interest- +ingly, Smax is symmetric with respect to a transforma- +tion λj → −λj which transforms ˜ρss +jl → −˜ρss +jl for all l ̸= j. +This can be intuitively explained by Smax only depending +on the norm of coherences. In this case, the phase prefer- +ence of entrainment and mutual coupling is reversed, i.e. +both prefer out-of-phase in the refrigerator regime while +mutual coupling (entrainment) prefers in-phase (out-of- +phase). +Scaling with N.– Calculating Smax boils down to per- +forming N-variable optimization which in general is dif- +ficult for N > 2. However, in the refrigerator regime, +assuming homogeneous driving λj = λ the problem sim- +plifies and one can show that S({ϕ1j}) saturates the l1- +norm bound [37] (see Append. A for a proof). Thus, we +conclude that in the refrigerator regime Smax ∝ Cl1 = +�N+1 +inh +γc(nc − nh) +8nh[γc(1 + nc) + γh(1 + nh)]. +(13) +The asymptotic scaled Smax above only depends on the +bath properties and is independent of the drive strength. +Furthermore, as shown in Fig. 3b Smax follows a sub- +linear power law behavior and all the optimum phases +{ϕj1}|S=Smax coalesce to a single phase 3π/2 (Fig. 3d). +In the engine case, it is difficult to find an analytic +closed-form expression for Smax. +However, we numer- +ically observe in Fig. 3a, that the competition between +entrainment and mutual coupling persists for any N since +Smax is smaller than entrainment contribution causing +Smax → 0. This decay is due to phase repulsiveness be- +cause of mutual coupling as shown in 3c. Thus, in the +large N-limit, the qualitative behavior of this model is +analogous to the Kuramoto model with phase-repulsive +coupling, where the mean-field synchronization order pa- +rameter approaches zero [47]. +Summary.– We have shown that there exists an interplay +between entrainment and mutual coupling in a collec- +tively driven-dissipative degenerate thermal maser. The +interplay depends on the thermodynamic functionality of +the maser, i.e., they compete in the engine regime and +cooperate in the refrigerator regime. The results rely on +two key ingredients: i. a coherent drive that collectively +couples to the degenerate manifold causing entrainment +and mutual coupling to coexist and ii. a dissipative mech- +anism that causes a population inversion between the +non-degenerated and degenerated manifolds to observe +the competition. +We demonstrate our findings using a minimal model +of a generalized Scovil–Schulz-DuBois maser heat engine +and show that in the thermodynamic limit (N → ∞) the +dominance of mutual coupling leads to phase repulsive- +ness causing the engine’ working substance to be asyn- +chronized (Smax = 0). On the other hand, since there is +cooperation in the refrigerator case, the phases coalesce +to 3π/2 giving a finite Smax that is independent of system +properties. In other words, as the system size increases +in order for the working substance to be synchronized the +external drive needs to perform work on the system. +Our work not only contributes to the growing field +of quantum synchronization by adding valuable insights +when distinct synchronizing mechanisms coexist but +helps understand quantum heat engines from a synchro- +nization perspective. +Acknowledgments.– +This +research +was +supported +by +the +Institute +for +Basic +Science +in +South +Ko- +rea (IBS-R024-Y2). +S.V. acknowledges support from +a Government of India DST-QUEST grant number +DST/ICPS/QuST/Theme-4/2019. +The authors would +like to thank V. Singh for the useful discussions. +∗ sai@phy.iitb.ac.in +† juzar˙thingna@uml.edu +[1] A. Pikovsky, M. Rosenblum, and J. Kurths, Synchroniza- +tion: a universal concept in nonlinear science (American +Association of Physics Teachers, 2002). +[2] R. Adler, Proceedings of the IRE 34, 351 (1946). +[3] J. A. Acebr´on, L. L. Bonilla, C. 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Solanki, F. M. Mehdi, M. Hajduˇsek, and S. Vinjanam- +pathy, arXiv preprint arXiv:2212.09388 (2022). +[47] L. S. Tsimring, N. F. Rulkov, M. L. Larsen, and M. Gab- +bay, Phys. Rev. Lett. 95, 014101 (2005). +[48] T. Baumgratz, M. Cramer, and M. B. Plenio, Phys. Rev. +Lett. 113, 140401 (2014). +Appendix A: Synchronization in D-level system +Here, we will derive the formula for the synchronization measure Smax for a D-level system from the Husimi-Q +representation +Q[ρ] = +D! +πD−1 ⟨αD|ρ|αD⟩ , +(14) +where |αD⟩ = �D +n=1 αn |n⟩ is the SU(D) coherent state [40] with +αn = +� +eiφn cos θn +�n−1 +k=1 sin θk +1 ≤ n < D +eiφD �D−1 +k=1 sin θk +n = D. +(15) +It has been implicitly assumed that for n = 1 the product term is just an identity. We are only concerned with +the distribution of phases, so we can integrate out the polar angles dΘ = �D−1 +l=1 cos θl(sin θl)2D−2l−1dθl from Q[ρ] to +obtain a quasi-probability distribution over a D − 1 torus, i.e., +� π/2 +0 +Q[ρ] dΘ = +D! +πD−1 +D−1 +� +n,m=1 +ρnm +� π/2 +0 +α∗ +nαm +D−1 +� +l=1 +cos θl (sin θl)2D−2l−1 dθl. +(16) +The diagonal contribution (n = m) gives, +� π/2 +0 +|αn|2 +D−1 +� +l=1 +cos θl(sin θl)2D−2l−1 dθl = +1 +2D−1D! +∀n = 1, · · · , D, +(17) +while the off-diagonal (n ̸= m) contribution yields, +� π/2 +0 +α∗ +nαm +D−1 +� +l=1 +cos θn(sin θl)2D−2l−1 dθl = +π +2D+1D!ei(φm−φn) +∀n ̸= m = 1, . . . , D. +(18) +Therefore, combining the diagonal and the off-diagonal contributions we obtain, +� π/2 +0 +Q[ρ] dΘ = +1 +(2π)D−1 + +1 +2D+1πD−2 +� +n̸=m +ρnmei(φm−φn). +(19) + +7 +The first term represents the contribution from a uniform distribution, which can be eliminated by defining the phase +quasi-probability distribution +S(φ1, . . . , φD−1) := +� π/2 +0 +Q[ρ] dΘ − +1 +(2π)D−1 = +1 +2D+1πD−2 +D +� +n̸=m +ρnmei(φm−φn). +(20) +The synchronization measure Smax in the main text is simply the maximum of Eq. (20), i.e., +Smax ≡ +max +φ1,··· ,φD−1 S(φ1, · · · , φD−1) ≤ +1 +2DπD−2 Cl1, +(21) +where Cl1 = � +n nh). In this case, we have arg(ρ1j) = π/2 and arg(ρjl) = 0 (j ̸= l) and by using the +steady-state solutions [Eqs. (38) and (42)] in the phase quasi-probability distribution, Eq. (20), we obtain, +Smax|nc>nh = +1 +2N+2πN max +{ϕ1j} +� N+1 +� +j=2 +|˜ρss +1j| sin ϕj1 − +N+1 +� +jnh = +1 +(2π)N +λ2γc(nc − nh)(N 2 + (2k − 1)N) +8F(N, λ, γc, γh, nh, nc) +, +(48) +where k = γh(1 + nh)/λ = |ρss +1j|/|ρss +jk| is the dissipation-to-driving ratio. In the limit of macroscopic degeneracy +N → ∞, (2π)NSmax approaches a constant value given in Eq. (13) of the main text. +In the engine case (nc < nh), the optimization is trickier. In this regime, we have competition between entrain- +ment and mutual coupling as can be seen from arg(ρ1j) = −π/2 and arg(ρjl) = π [see Eqs. (38) and (42)]. The +synchronization measure Smax can be expressed as, +Smax|nh>nc = +1 +2N+2πN max +{ϕ1j} +� N+1 +� +j=2 +|˜ρss +1j| sin ϕj1 − +N+1 +� +jnc = +1 +16π2 |˜ρss +23| max +ϕ21,ϕ31 +� +k sin ϕ21 + k sin ϕ31 − cos(ϕ31 − ϕ21) +� +. +(50) +Thus, calculating Smax is now reduced to optimizing a two-variable function f(x, y) ≡ k sin x + k sin y − cos(x − y). +One can easily verify +max +x,y f(x, y) = +� +� +� +2k − 1 +if k ≥ 2 +1 + k2 +2 +if 0 ≤ k ≤ 2, +(51) +with optimum points (x, y) = (π/2, π/2) when k ≥ 2 and {(arcsin(k/2), π−arcsin(k/2)), (π−arcsin(k/2), arcsin(k/2))} +when k ≤ 2. By substituting Eq. (51) in Eq. (50), one obtains Eq. (11) of the main text. +Appendix D: Smax = 0 if and only if ρ is diagonal (D = 3) +Next, we will show that in the case of D = 3, Smax = 0 if and only if ρ is diagonal. We consider a three-level system +with {|0⟩ , |1⟩ , |2⟩} representing the eigenvectors. A general expression for S ≡ S(φ0, φ1, φ2) for such a three-level +system reads, +S = 1 +8π [|ρ01| cos(φ1 − φ0 + Φ01) + |ρ02| cos(φ2 − φ0 + Φ02) + |ρ12| cos(φ2 − φ1 + Φ12)], +(52) +where Φij = arg(ρij). We first transform the equation by defining ϕij = φi − φj, i.e., +S = 1 +8π [|ρ01| cos(ϕ10 + Φ01) + |ρ02| cos(ϕ20 + Φ02) + |ρ12| cos(ϕ20 − ϕ10 + Φ12)]. +(53) +Given that the reduced density matrix ρ is diagonal, S(ϕ10, ϕ20) is zero everywhere (∵ |ρij| = 0 ∀i, j) and thus it is +trivial that Smax = 0. +However, it is not trivial to show that if Smax = 0 the ρ will be diagonal. We will prove it by contradiction. +Let us assume ρ is not diagonal and Smax = 0. Then, by definition S(ϕ10, ϕ20) is zero or negative everywhere else. +We will show below, considering all possible cases, that we can always find {ϕ10, ϕ20} such that S is positive, ergo +contradiction. Case 1: Only one coherence is non-zero, let’s say ρ01. Then, we can choose ϕ10 = −Φ01 such that +S = |ρ01| > 0. +Case 2: Two coherences are non-zero, let’s say ρ01 and ρ02. We can then choose ϕ10 = −Φ01 and ϕ20 = −Φ02 such +that S = |ρ01| + |ρ02| > 0 +Case 3: All coherences are non-zero. This is a non-trivial case. First, let us choose (ϕ10, ϕ20) = (π/2−Φ01, π/2−Φ02) +such that +S = 1 +8π |ρ12| cos(Φ01 − Φ02 + Φ12) > 0. +(54) + +10 +The above is positive if the cosine term is positive. If it is negative, we can just choose (ϕ10, ϕ20) = (−π/2−Φ01, π/2− +Φ02) such that S remains positive, i.e., +S = − 1 +8π |ρ12| cos(Φ01 − Φ02 + Φ12) > 0. +(55) +If the cosine term is zero, we choose (ϕ10, ϕ20) = (−Φ01, −Φ02) to keep S positive, +S = 1 +8π (|ρ01| + |ρ02|) > 0. +(56) +Thus, we conclude that in all cases, we can always find phases configuration such that S is positive, implying that +Smax cannot be zero if ρ is not diagonal. + diff --git a/etE3T4oBgHgl3EQfHQmJ/content/tmp_files/load_file.txt b/etE3T4oBgHgl3EQfHQmJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e45329f3470ff5f2697d5f7b088ba8ec4ea43c8a --- /dev/null +++ b/etE3T4oBgHgl3EQfHQmJ/content/tmp_files/load_file.txt @@ -0,0 +1,692 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf,len=691 +page_content='Cooperation and Competition in Synchronous Open Quantum Systems Taufiq Murtadho,1, 2, 3 Sai Vinjanampathy,4, 5, 6, ∗ and Juzar Thingna1, 2, 7, † 1Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon 34126, Republic of Korea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' 2Basic Science Program, Korea University of Science and Technology, Daejeon 34113, Republic of Korea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' 3School of Physical and Mathematical Science, Nanyang Technological University, Singapore 639798, Singapore 4Department of Physics, Indian Institute of Technology-Bombay, Powai, Mumbai 400076, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' 5Centre of Excellence in Quantum Information, Computation, Science and Technology, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' 6Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' 7Department of Physics and Applied Physics, University of Massachusetts, Lowell, MA 01854, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' (Dated: January 12, 2023) Synchronization between limit cycle oscillators can arise through entrainment to an external drive or through mutual coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' The interplay between the two mechanisms has been studied in clas- sical synchronizing systems, but not in quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Here, we point out that competition and cooperation between the two mechanisms can occur due to phase pulling and phase repulsion in quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' We study their interplay in collectively driven degenerate quantum thermal ma- chines and show that these mechanisms either cooperate or compete depending on the working mode of the machine (refrigerator or engine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' The entrainment-mutual synchronization interplay persists with an increase in the number of degenerate levels, while in the thermodynamic limit of degener- acy, mutual synchronization dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Overall, our work investigates the effect of degeneracy and multilevel scaling of quantum synchronization and shows how different synchronizing mechanisms can cooperate and compete in quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content='– Synchronization is a ubiquitous phe- nomenon in which stable phase relations emerge between multiple limit cycle oscillators [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' There are two main mechanisms that give rise to synchronization: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' En- trainment that refers to synchronization of an oscillator by unidirectional coupling to a periodic external drive [2], and ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Mutual synchronization which refers to the adjustment of rhythms of two or more mutually coupled oscillators, such as in the widely-known Kuramoto model [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' These two mechanisms may coexist in some systems [4–7] and their interplay has also been experimentally studied in globally coupled electrochemical oscillators [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' In the same spirit as classical synchronization, quan- tum synchronization is often studied through entrain- ment [9–14] or mutual coupling [15–21] and has been experimentally observed recently [22–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' However, un- like classical synchronization, the coexistence and the in- terplay between these two mechanisms in the quantum regime has not been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Understanding this in- terplay is crucial in the control of various quantum tech- nologies where both driving and interaction are impor- tant such as in superradiant lasers [16], coupled time- crystals [25], and coupled heat engines [26–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' In this work, we show that the phases of steady-state coherences follow a phase synchronization model, where the external entraining drive competes with the mu- tually coupled phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' This opens up the possibility of observing well-studied classical phenomena, such as synchronization-anti-synchronization transition [30] and chimera [31, 32], in the quantum regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Our framework applies to generic quantum systems, with one or more ex- ternal drives that couple the coherences that themselves are mutually coupled, either coherently or dissipatively, leading to an interplay between entrainment and mutual synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' As a concrete example, we consider a degenerate mul- tilevel generalization of the Scovil–Schulz-DuBois maser heat engine [33], where the external collective drive con- nects transitions between the degenerate manifold and the first-excited state [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' The states within the degen- erate manifold mutually interact to form a stable col- lective symmetric (in-phase) and anti-symmetric (out-of- phase) superposition (mutual synchronization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' At the same time, the external drive causes the phases within the degenerate manifold to be aligned in-phase with the drive (entrainment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' In the engine regime, stimulated emission consumes the collective symmetric superposi- tion state thereby enhancing the population of the anti- symmetric state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Thus, there is competition between en- trainment (in-phase) and mutual synchronization (out- of-phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' In the refrigerator regime, the stimulated ab- sorption enhances the population of the collective sym- metric superposition state thereby always cooperating with entrainment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Our work sheds light on the synergis- tic interplay between entrainment and mutual synchro- nization in quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Quantum synchronization in D-level systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content='– Quan- tum synchronization has been studied in systems with continuous degrees of freedom such as oscillators [9– 11, 13, 15, 17, 35] and discrete degrees of freedom such as spin-1 systems [12, 14, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' A wide variety of mea- sures, based on various physical and mathematical mo- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content='04322v1 [quant-ph] 11 Jan 2023 2 tivations such as phase-space based measures [9, 12, 20], correlation measures [36], and information-theoretic mea- sures [37] has been used to quantify synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' In this work, we use the phase-space based measure built on the Husimi-Q phase space representation [38, 39] of the steady-state ρss with respect to SU(D) coherent state [39, 40] defined as Q[ρss] = D!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' πD−1 ⟨αD|ρss|αD⟩ , (1) where |αD⟩ = �D n=1 αn |n⟩ is the SU(D) coherent state with coefficients αn = � eiφn cos θn �n−1 k=1 sin θk 1 ≤ n < D eiφD �D−1 k=1 sin θk n = D, (2) where it is implicitly assumed that the product term is identity for n = 1 and the reference phase φ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' The synchronization measure is given by the difference be- tween integrating out the angles θk corresponding to the population degrees of freedom and doing the same for the uniform measure, given by S(φ1, · · · , φD−1) = � Q[ρss]dΘ − 1 (2π)D−1 = 1 2D+1πD−2 � n̸=m ρss nmei(φm−φn), (3) which lives on a D − 1 dimensional torus (see Append.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' The distribution S(φ1, · · · , φD−1) is zero everywhere for a diagonal steady-state which is interpreted as a limit cycle [37] possessing stable amplitudes (fix diagonal ele- ments) but free phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' The notion of free-phase in a such diagonal limit cycle is analogous to a classical stochastic limit cycle whose phase distribution approaches a uni- form distribution in the steady-state [1, 13, 14, 41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' We associate the peak of S(φ1, · · · , φD−1) as a phase- space synchronization measure [12, 20, 42], Smax = max φ1,··· ,φD−1 1 2D+1πD−2 � n̸=m ρss nmei(φm−φn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' (4) The synchronization measure, Smax only depends on the steady-state coherences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' However, we note that a high value of Smax requires all phase preferences Φij = arg(ρss ij ) to be compatible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=', Φij − Φjk = Φik ∀i ̸= j ̸= k, a condition that is stronger than the mere presence of coherences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Degenerate thermal maser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content='– Entrainment in quantum systems is the result of an interplay between coherent driving and dissipation [10, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' The system we consider is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' and consists of (N + 2) levels whose bare Hamiltonian is given by, H0 = ω1 |1⟩ ⟨1| + N+1 � j=2 ωj |j⟩ ⟨j| , (5) |0⟩ |1⟩ |2⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' |N + 1⟩ Th Tc λ, Ω ∆ ω2 ω1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Schematic of the degenerate quantum thermal maser, which is a generalization of the standard Scovil– Schulz-DuBois three-level thermal maser [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Here, N is the number of states in the degenerate manifold and here we focus on the case ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' The near-degenerate case where ∆ ̸= 0 is discussed in the accompanying manuscript [43] with ωj+1 > ωj, ω0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' The upper N levels are degen- erate with ω2 = ω3 = · · · = ωN+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Although we work in the limit of exact degeneracy, our main results hold even in the near-degenerate scenario and will be considered in detail in an accompanying Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' This system is driven by a monochromatic drive V (t) of frequency Ω given by V (t) = N+1 � j=2 λjeiΩt |1⟩ ⟨j| + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' (6) This drive can be rewritten as a coupling to a collective- transition mode |1⟩ ↔ |J⟩ = (1/λeff) � j λj |j⟩ with λeff = �� j |λj|2 being the effective coupling strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Such collective drives are realizable in an ensemble of atoms driven by light, if the inter-atomic distance is much smaller than the wavelength of the light, such as in the case of Dicke superradiance [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' The system is acted upon by a dissipator D[ρ] = 2 � µ=1 � ΓcµL[cµ]ρ + N+1 � j=2 ΓhµL[hj µ]ρ � , (7) which leads to a multilevel generalization of the Scovil– Schulz-DuBois maser heat engine [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' The dissipator L[X]ρ = 2XρX†−{X†X, ρ} is of the Lindblad form such that the hot (cold) bath with jump operators hj 1 = hj† 2 = |0⟩ ⟨j| (c1 = c† 2 = |0⟩ ⟨1|) induce transitions between the ground state and the degenerated manifold (first-excited state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' The associated rates follow local-detailed balance and are given by Γh1(c1) = γh(c)(1 + nh(c)) and Γh2(c2) = γh(c)nh(c) with γh(c) being the effective system-bath cou- pling strength and nh(c) = � exp(βh(c)ω2(1)) − 1 �−1 be- ing the Bose-Einstein distribution at inverse temperature 3 βh(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' The action of the heat baths leads to a population inverted steady state between the first-excited state |1⟩ and the degenerated manifold {|j⟩, ∀j = 2, · · · , N + 1} if nh > nc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' If there is population inversion, the system behaves as a maser heat engine [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' However, if nh < nc, population inversion is lost and the system behaves as a refrigerator by attenuating the drive [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' We can rewrite the Hamiltonian in a frame co-rotating with the drive as ˜H = (Ω/2)(�N+1 j=2 |j⟩ ⟨j| − |1⟩ ⟨1|) giving us the rotating frame quantum master equation, d˜ρ dt = −i[H0 − ˜H + ˜V , ˜ρ] + D[˜ρ], (8) where ˜O ≡ e−i ˜ HtOei ˜ Ht (O = ρ, V ) is an operator in the rotated frame with ˜V = �N+1 j=2 λj |1⟩ ⟨j| + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content='. Competition vs cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content='– Equation (8) can be solved analytically for the case of homogeneous driving strength λj = λ (∀j = 2, · · · , N + 1) and resonant driving Ω = ω2 − ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' In this case, the steady-state coherences are given by ˜ρss 1j = iλ(nc − nh)γcγh(1 + nh) F(N, nh, nc, γc, γh, λ) , (9) ˜ρss jl = λ2γc(nc − nh) F(N, nh, nc, γh, γc, λ), (10) where j, l = 2, · · · , N + 1, j ̸= l and the function F(N, nh, nc, γc, γh, λ) = AN 2 + BN + C with A, B, and C being positive constants that depend on all remaining parameters (see Append.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' B for the explicit expressions for these constants).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' The non-degenerate coherences (˜ρ1j) are directly in- duced (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=', ∝ λ) by the drive whereas the degenerate co- herences (˜ρjl) are an indirect consequence (∝ λ2) of the collective nature of the drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Their differences are clear as one transforms back to the original frame in which ρ1j = ˜ρ1je−iΩt and ρjl = ˜ρjl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' The phase preferences induced by ρ1j rotate with the driving frequency while that of ρjl remain stationary in the original frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Both of these coherences affect the phase distributions of the states within the degenerate manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' For these reasons, we infer that there are two synchronization mechanisms at play in this system, entrainment induced directly by the drive and mutual coupling that occurs due to the presence of a degenerate manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Entrainment induces phases relative to driving whose effect is the emergence of stable non-degenerate coherences ˜ρss 1j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' On the other hand, mutual coupling induces a relative phase between states in the degenerated manifold independent of the driving phase, which is reflected by stable degenerate co- herences ˜ρss jl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Recall that we have denoted Φij = arg(˜ρss ij ) as the steady-state phase preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' When there are multi- ple of such preferences, synchronization requires all the phase relations to be compatible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Φij − Φjk = Φik (i ̸= j ̸= k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' However, we find that in our system such a a −π − π 2 0 π 2 π −π − π 2 0 π 2 π ϕ31 ϕ21 −5 · 10−4 5 · 10−4 S(ϕ21, ϕ31) b −π − π 2 0 π 2 π −π − π 2 0 π 2 π ϕ31 ϕ21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content='5 2 0 1 2 c nh/nc Smax × 10−4 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content='5 1 1 2 3 d λ2/λ3 Smax × 10−4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Interplay between entrainment and mutual coupling for N = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Panels a and b show phase quasi-distribution func- tion S(ϕ21, ϕ31) [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' (3)] where ϕij = φi − φj in the engine regime (nh/nc = 100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' For k = 3, S(ϕ21, ϕ31) shows a local- ized maximum when the phases are in-phase (ϕ21 − ϕ31 ≈ 0 in the red-region in a, entrainment-dominant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Whereas for k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content='75, when S(ϕ21, ϕ31) is maximized the phases do not localize but their difference is out-of-phase (ϕ21 − ϕ31 ≈ π in the red-region in b, mutual coupling dominant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Panel c shows Smax (solid circle) as a function of nh/nc with the solid line representing the analytic prediction of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' The dashed line is the entrainment contribution to Smax, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=', (|ρ12| + |ρ13|)/16π2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' The vertical dotted line represents the boundary between refrigerator (nh/nc < 1) and engine (nh/nc > 1) regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Panel d shows Smax (solid circle) and (|ρ12| + |ρ13|)/16π2 (dashed line) plotted against inhomoge- neous driving strength ratio |λ2/λ3| ≤ 1 in the engine (red) and refrigerator (blue) regimes indicating competition (coop- eration) between entrainment and mutual coupling is robust in the engine (refrigerator) regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' The other parameter val- ues are ω2 = ω3 = 3ω1, Ω = ω2 − ω1, γc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content='2ω1, γh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content='05ω1, nc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content='5, and λ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content='1ω1 condition is only satisfied in the refrigerator regime where Φ1j = π/2 (∀j) and Φjl = 0 (j ̸= l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' In the engine regime, we have Φ1j = −π/2 (∀j) and yet Φjl = π (j ̸= l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' We interpret this as a result of an interplay between entrain- ment and mutual coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' We find that entrainment always pulls the degenerate states to be in-phase (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Mutual coupling prefers out-of-phase configuration in the engine regime (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' 2b), and in-phase configura- tion in the refrigerator regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Consequently, we expect entrainment and mutual coupling to cooperate in the re- frigerator regime and compete in the engine regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' The competition and cooperation are obvious when we calculate the phase space synchronization measure Smax [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' (4)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' In general, this requires optimization over 4 2 8 14 20 2 4 ×10−2 a N (2π)NSmax 2 8 14 20 5 10 15 20 ×10−2 b N (2π)NSmax 0 π 3π 2 π 2 c 0 π 3π 2 π 2 d FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Panels a-b show Smax = (2π)NSmax (solid cir- cle) compared with its entrainment contribution 1 4 �N+1 j=2 |ρ1j| (empty circle) as a function of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' The error bar is calcu- lated from 102 random realizations of driving strength ratio λj/λ2 ≤ 1 for λj ≥ 0 and j = 3, · · · , N + 1 with λ2 held constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' The solid lines are the curve fits using Smax ∝ N α with α = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content='72 (a) and = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content='69 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Panels c-d show optimum phases {ϕopt j1 } for N = 20 in in the engine (c, nh/nc = 10) and refrigerator (d, nh/nc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content='4) regimes plotted on a unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' The different opacity represents different realizations of λj/λ2 (j = 3, · · · , N + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' All the phases in all realizations coalesce to a single data point in the refrigerator case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' All other parameters are the same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' N variables which we calculate analytically for N = 2 (see Append.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' C) Smax = 1 16π2 × � � � � � � � � � |˜ρss 12| + |˜ρss 13| + |˜ρss 23| if nhnc & k>2 � 1 + k2 2 � |˜ρss 23| if nh>nc & k<2, (11) where k = γh(1 + nh)/λ = |˜ρss 12|/|˜ρss 23| = |˜ρss 13|/|˜ρss 23| is the dissipation-to-driving ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' The set of optimal phases (ϕopt 21 , ϕopt 31 ) ≡ (ϕ21, ϕ31)|S=Smax evaluated in Append.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' C are given by, (ϕopt 21 , ϕopt 31 ) = � � � � � � � � −π 2 , −π 2 � if nhnc & k>2 (χ, π − χ) & (π − χ, χ) if nh>nc & k<2, (12) where ϕij = φi − φj and χ = arcsin(k/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Equations (11)-(12) show the effect of the coherent drive and bath couplings on the synchronous dynamics of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Cooperation in the refrigerator regime (nc > nh) is re- flected by the fact that each component of the magni- tude of coherence adds up in the synchronization mea- sure Smax, whereas in the engine case there is competi- tion since the mutual coupling component |ρss 23| reduces the effect of the entrainment contribution |ρss 12|+|ρss 13|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' In other words, the phases are either equal in some cases or they are arranged antipodally in other cases, as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' In the engine regime, Smax is also divided into regimes where entrainment is dominant (k > 2) and where the mutual coupling is dominant (k < 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' For the entrain- ment dominant regime, the competition is apparent from the negative contribution of |ρss 23| to Smax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Note that this is different from the previously reported phenomenon of synchronization blockade [14, 46], in our case, Smax can not vanish except for λ = 0 or nh = nc where the steady-state is diagonal (see Append.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' The transition from entrainment to mutual coupling dominant regime is shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' 2a-b where we plot the phase distribu- tion S(ϕ21, ϕ31) for different k values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' In particular, we see that as we cross k = 2, the relative phases go from in-phase to out-of-phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Moreover, the localization pat- tern changes from a point localization to ring localization (on a torus), wherein the latter only the relative phase ϕ23 = ϕ21 − ϕ31 is fixed, indicating that entrainment is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' The competition and cooperation observed is also ro- bust with respect to all values of individual driving strength ratio λ2/λ3 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Interest- ingly, Smax is symmetric with respect to a transforma- tion λj → −λj which transforms ˜ρss jl → −˜ρss jl for all l ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' This can be intuitively explained by Smax only depending on the norm of coherences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' In this case, the phase prefer- ence of entrainment and mutual coupling is reversed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' both prefer out-of-phase in the refrigerator regime while mutual coupling (entrainment) prefers in-phase (out-of- phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Scaling with N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content='– Calculating Smax boils down to per- forming N-variable optimization which in general is dif- ficult for N > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' However, in the refrigerator regime, assuming homogeneous driving λj = λ the problem sim- plifies and one can show that S({ϕ1j}) saturates the l1- norm bound [37] (see Append.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' A for a proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQfHQmJ/content/2301.04322v1.pdf'} +page_content=' Thus, we conclude that in the refrigerator regime Smax ∝ Cl1 = �N+1 i